Active Solar Collectors and Their Applications ARI RABL Center for Energy and Environmental Studies Princeton Universitv
New York Oxford OXFORD UNIVERSITY PRESS 1985
Active Solar Collectors and Their Applications
ARI RABL
OXFORD UNIVERSITY PRESS
Active Solar Collectors and Their Applications
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Active Solar Collectors and Their Applications ARI RABL Center for Energy and Environmental Studies Princeton Universitv
New York Oxford OXFORD UNIVERSITY PRESS 1985
Oxford University Press Oxford New York Toronto Delhi Bombay Calcutta Madras Karachi Kuala Lumpur Singapore Hong Kong Tokyo Nairobi Dar es Salaam Cape Town Melbourne Auckland and associated companies in Beirut Berlin Ibadan Mexico City
Nicosia
Copyright © 1985 by Oxford University Press, Inc. Published by Oxford University Press, Inc., 200 Madison Avenue, New York, New York 10016 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, pholocopying, recording, or otherwise, without the prior permission of Oxford University Press. Library of Congress Cataloging in Publication Data Rabl, Ari. Active solar collectors and their applications. Includes index. 1. Solar collectors. TJ812R33 1985 ISBN 0-19-503546-1
Printing (last digit):
1. Title. 621.47
987654321
84-14861
PREFACE
The field of solar energy has grown too large to receive full coverage in a single book. The division into active and passive systems is convenient. Even though the distinction is not always clear-cut, one can say basically that active systems use collectors and storage as separate components with pumped energy transport, while passive systems rely on natural energy transfer. The past decade has seen tremendous progress in active solar technology. Many new collector types have been built and tested, and great strides have been made in the development of high temperature systems. An extensive program of demonstration projects, from domestic water heaters to central station power plants, has given us a large amount of operating experience. The insolation data base has been improved and expanded, and new powerful tools have been developed for the analysis of solar energy systems. In this book I have tried to summarize these developments into a form that can serve both as general reference and as text for solar energy courses. Technological progress is characterized by a learning curve; with time we find ways to do things better, cheaper and faster. Such learning can be discerned not only in the technology itself but also in the methods of explaining the technology. In a few textbook pages we now condense information that used to required countless volumes of journals and reports. I have consolidated a large amount of information by following a unified approach in the analysis of all collector types. For example, some elegant methods have been developed for the analysis of flat plate collectors and residential solar applications: the heat transfer factors of Hottel, Whillier, Bliss, and de Winter, the utilizability of Liu and Jordan, the f-chart of the University of Wisconsin, and others. Yet few people have taken advantage of these methods when discussing concentrating collectors or nonresidential applications. This is the first book to exploit the full potential of a unified presentation of active solar technologies. As much as possible I have pointed out the general applicability of the basic formulas, and I have illustrated them with a wide range of examples. Some of these include photovoltaics, because the methods can be used for the performance analysis of photovoltaic as well as thermal systems. In solar energy calculations one usually finds that the effort required increases rapidly with the desired accuracy. Frequently, the basic input is so
vi
Preface
uncertain that a simple approximation provides as good an answer as one could hope for. In other situations one may want to evaluate small differences between different systems, and a more precise analysis is needed. Accordingly I have presented the full spectrum of methods, from simple to complex, in order to allow the reader to select the method with the appropriate level of accuracy for each application. I believe that the book is complete in the sense of providing the tools for analyzing any active solar collector or system that might be practical. The fundamentals of solar energy are covered in Chapters 1 to 4: Chapter 1 gives an overview of solar technologies, Chapter 2 explains solar geometry, Chapter 3 deals with insolation data and models, and Chapter 4 discusses the basic properties of solar collectors. Chapters 5 to 8 treat the optics, and Chapters 9 and 10, the heat transfer of solar collectors. System analysis and applications form the subject of the next three chapters, including one on practical problems. The book ends with economic analysis and optimization. By selecting appropriate sections one can use this book as a text for a oneor a two-semester course, at the undergraduate or graduate level. For example, I have been teaching a one-semester junior level course, using most but not all of Chapters 1 to 4, 11, 13, and 14, as well as parts of Chapters 5, 6, 7, and 12. Finally, I come to the acknowledgments. This task is as pleasant as it is difficult to do justice to the countless individuals who have helped me in this project, in some form or other, directly or indirectly. I owe a debt of gratitude to all of them, and I beg for understanding and forgiveness if I fail to acknowledge anybody's contribution adequately. This book has benefited from innumerable conversations, discussions, and correspondence with colleagues both in the United States and abroad. My thanks to the people who have sent me material for possible inclusion in the book. As far as citations of literature are concerned, I have come to appreciate the difficulty of including each and every relevant reference. Let me say that 1 have tried to be fair to all. If I have overlooked a reference, it was probably an oversight or lack of awareness. Margaret Steinbugler collected most of the material for Chapter 13 with her senior thesis. The students of Engineering 319 were exposed to the intial versions of the manuscript, and their suggestions and comments are appreciated. Pat Eicker, Jim Freese, George Eggers, Donna Hawkins, George Kirk, Carl Lampert, Jim Leonard, Bill Schertz, and Gregory Tranter kindly supplied photos or glossy prints. A critical part of book writing is the review process, and I am much obliged to the people who patiently read through parts of the draft and made valuable suggestions: Frank Biggs, Dimitry Gondikas, Manuel Collares-Pereira, Bill Dickinson, David Faiman, David Govaer, Sandy Klein, Frank Kreith, Lorin Vant-Hull, Roland Winston, Rob Socolow, and Yair Zarmi.
Preface
vii
The most important readers have been Jeff Gordon and Jan Kreider, who gave me thorough reviews of the entire manuscript. I am thankful to my friends and colleagues who encouraged me during the course of writing this book, and I would like to express my gratitude to the people who provided the institutional framework and support for this enterprise: Frank Kreith and Ken Touryan at SERI and Rob Socolow at Princeton. My most deeply felt acknowledgment goes to the colleagues with whom I have collaborated over the years, above all: Carl Nielsen at Ohio State University; Veronika Rabl at Argonne National Laboratory; Manuel CollaresPereira, Joe O'Gallagher, and Roland Winston at the University of Chicago; Paul Bendt and Frank Kreith at SERI; David Faiman, Jeff Gordon, and Yair Zarmi at the Institute of Desert Research in Israel; and Hal Feiveson, Rob Socolow, Bob Williams, and Frank von Hippel at Princeton University. Working with such stimulating people has been a joy and a privilege. Princeton,N.J. July 1984
A. R.
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CONTENTS
1 SURVEY OF SOLAR EQUIPMENT AND APPLICATIONS 1.1 The Promise and the Challenge of Solar Energy 1.2 Survey of Applications 5 1.3 Solar Collector Technology 8 1.3.1 Flat Plates and Solar Ponds 11 1.3.2 Evacuated Tubes 13 1.3.3 Concentrating Collectors 16 1.4 Thermal Storage Technologies 23 1.5 Storage, Backup, and the Electric Utilities 24 References 26 2 SOLAR GEOMETRY 2.1 2.2 2.3 2.4 2.5
28
The Source of Radiation 28 Sun-Earth Geometry 29 Incidence Angles on Fixed Surfaces 32 Incidence Angles on Tracking Surfaces 37 Projected Incidence Angles 39 References 46
3 INSOLATION DATA AND MODELS 3.1 3.2 3.3 3.4 3.5
3
48
Solar Spectrum 48 Types of Insolation 49 Instruments for Measuring Solar Radiation 52 Insolation Data 54 Insolation Models 59 3.5.1 Types of Models 59 3.5.2 Clear Sky Radiation 62 3.5.3 Correlations between Beam, Diffuse, and Hemispherical Radiation 64 3.6 Long Term Average Insolation 66 3.7 Frequency Distribution of Daily Radiation Values 71 3.8 Yearly Irradiation on Aperture 73
3
x
Contents
3.9 Seasonal Variation of Insolation 3.10 Summary 79 References 80
75
4 COLLECTOR EFFICIENCY AND COLLECTOR TESTING 4.1 Definition of Instantaneous Collector Efficiency 4.2 4.3 4.4 4.5
4.1.1 Specification of Insolation 84 4.1.2 Optical Efficiency and U Value
83
83
86
Efficiency Based on Fluid Temperature 90 Incidence Angle Modifiers 96 Test Results 99 Standard Collector Test Procedures 101
4.5.1 Instantaneous Efficiency 4.5.2 Collector Time Constant
101 106
4.6 Simplified Collector Test Procedures
108
4.6.1 Measurement of Heat Loss 109 4.6.2 Warm-up Test 110 4.6.3 Masked Stagnation Test 110
4.7 Measuring the Optical Quality of Focusing Collectors References 112
111
5 FUNDAMENTALS OF OPTICS FOR SOLAR COLLECTIORS 114 5.1 Reflection
115
5.2 Refraction
117
5.1.1 The Law of Reflection 115 5.1.2 Optics of Troughlike Reflectors—Projection of Ray Trace Diagram 116
5.2.1 5.2.2 5.2.3 5.2.4 5.2.5 5.2.6
Snell's Law of Refraction 117 Fresnel Equations 118 Multiple Reflections 119 Absorption in the Cover 121 Absorption at the Absorber 122 Optics of Glass Tubes 124
5.3.1 5.3.2 5.3.3 5.3.4
Definition of Concentration Ratio 125 Thermodynamic Limit of Concentration 126 Acceptance of Diffuse Radiation 131 Operating Temperature as a Function of Concentration
5.4.1 5.4.2 5.4.3 5.4.4
Angular Distribution of Beam Radiation 133 Statistics of Reflecting Surfaces 136 Effect of Surface Errors on Width of Reflected Beam Combination of Optical Errors 141
5.3 Concentration of Solar Radiation
125
5.4 Image Spread Due to Finite Source and Optical Errors
5.5 Image Spread for Line Focus Collectors References 145
142
132
133 138
Contents
xi
6 OPTICS OF NONTRACKING COLLECTORS 6.1 6.2 6.3 6.4
147
Compound Parabolic Concentrators 147 Reflectors for Evacuated Tubes 157 V-Troughs 164 Side Reflectors 165 References 167
7 TRACKING CONCENTRATOR TYPES 7.1 Parabolic Reflectors 7.2 Fresnel Reflectors 7.2.1 7.2.2 7.2.3 7.2.4
170
170 173
Line-Focus and Point-Focus Central Receivers 173 Tracking Motion for Linear Fresnel Reflectors 175 Off-Axis Aberrations of Spherical Reflectors 177 Curvature of Heliostats for Central Receiver 180
7.3 Fresnel Lenses 182 7.4 Fixed Reflectors with Tracking Receivers
186
7.4.1 Spherical Reflector 186 7.4.2 Circular Cylindrical Reflector with Tracking Receiver 7.4.3 Reflector Slats on Circular Cylindrical Mount 188
7.5 Concentrator Configurations for Low-Cost Manufacture 7.6 Second Stage Concentrators 190 References 193
186
189
8 OPTICAL ANALYSIS AND OPTIMIZATION OF PARABOLIC REFLECTORS 196 8.1 Analytical Framework 196 8.2 Angular Acceptance Function 8.2.1 Parabolic Troughs 8.2.2 Parabolic Dishes
8.3 Effective Source
198 200
198
202
8.3.1 Effective Source for Line-Focus 8.3.2 Effective Source for Point-Focus
202 204
8.4 Flux at Receiver and Intercept Factor 205 8.5 Approximation of Sunshape by Gaussian Distribution 8.6 Intercept Factor and Circumsolar Radiation 209 References 211 9 HEAT TRNASFER IN SOLAR COLLECTORS 9.1 9.2 9.3 9.4 9.5
General Considerations 212 Heat Flow to Ambient 217 Planar Configuration 222 Tubular Configuration 225 Concentrator Configuration 227
212
206
xii
9.6 9.7 9.8 9.9
Contents
Heat Transfer Inside Tubes and Ducts 229 Conduction 231 Other Configurations 234 Heal Loss Coefficients of Typical Collectors 236
9.9.1 Evacuated Tubes 236 9.9.2 Tubular Receivers for Parabolic Troughs 9.9.3 Flat Plates 241
240
9.10 Warming of Glazing by Absorption of Solar Radiation References 248 10 HEAT TRANSFER FACTORS 10.1 10.2 10.3 10.4 10.5
251
Temperature Distribution in Collector Mean Fluid Temperature 252 Fluid Inlet Temperature 263 Heat Exchanger Factor 268 Collector Arrays and Pipe Loss Factors References 275
11 SYSTEM MODELS
251
273
277
11.1 System Configuration and Controls 277 11.2 System Performance Analysis 279 11.2.1 Computer Simulations 11.2.2 Shorthand Procedures
11.3 Yearly Collectible Energy 11.3.1 11.3.2 11.3.3 11.3.4
280 286
287
Radiation above a Specific Threshold Assumptions for Yearly Correlations Operating Time 305 Other Thresholds 307
11.4 Monthly Utilizability Method 11.5 The ,f-Chart 317 References 322 12 APPLICATIONS
290 300
309
325
12.1 Active Space Heating and Cooling
325
12.1.1 12.1.2 12.1.3 12.1.4
System Configurations 326 Performance Prediction for Space Heating Measured System Performance 330 Solar Cooling 330
12.2.1 12.2.2 12.2.3 12.2.4
System Types 332 Performance Comparison 335 Performance Prediction for Water Heaters Swimming Pool Heaters 337
12.2 Water Heating
327
332
336
246
Contents
xiii
12.3 Industrial Process Heat 338 12.3.1 General Considerations 338 12.3.2 Systems without Storage for Constant Loads 12.3.3 Single-Pass Open-Loop System with Storage 12.3.4 Closed-Loop Configurations 351 12.4 Central Heat Collection 354 12.5 Power Generation 356 12.5.1 Photovoltaics 357 12.5.2 Solar Thermal Power 359 References 361 13 PRACTICAL CONSIDERATIONS
364
13.1 Materials 364 13.1.1 Absorber 364 13.1.2 Cover 369 13.1.3 Reflector 373 13.1.4 Other Materials 377 13.2 Installation 381 13.2.1 Collector Location and Orientation 13.2.2 Pumps and Piping 384 13.2.3 Safety and Protection of Equipment 13.3 Maintenance and Cleaning 387 13.3.1 Maintenance 387 13.3.2 Cleaning 388 13.3.3 Tracking 392 References 393
14 ECONOMIC ANALYSIS
340 345
381 386
396
14.1 Constant Currency 396 14.2 Comparing Present and Future Costs 397 14.2.1 The Time Value of Money 397 14.2.2 How to Choose the Rate for Discounting the Future 14.2.3 Levelizing 400 14.2.4 Discrete and Continuous Cash Flows 405 14.3 Economic Evaluation Criteria 407 14.3.1 Life Cycle Savings 407 14.3.2 Internal Rate of Return 408 14.3.3 Cost of Saved Energy 409 14.3.4 Payback Period 410 14.3.5 Comparison of Criteria 415 14.4 Other Costs 416 14.5 Debt Financing, Taxes, and Inflating Currency 418 14.5.1 Principal and Interest 419 14.5.2 Depreciation 421
398
xiv
Contents 14.5.3 The Complete Formula 421 14.5.4 Constant Currency versus Inflating Currency References 424
15 SYSTEM OPTIMIZATION
426
15.1 Optimization Criteria 426 15.2 Optimization 427 15.2.1 Equations for Optimization 427 15.2.2 Photovoltaics with Sellback to Utility 429 15.2.3 Industrial Process Heat Systems 433 15.3 System Value 436 15.3.1 Definition of Value 436 15.3.2 Value of System Changes 437 15.4 Value of Storage 439 15.4.1 General Formulation 439 15.4.2 Value of Storage for Industrial Process Heat References 444 Appendix A Nomenclature Appendix B Units
445
450
Appendix C Properties of Materials Appendix D Meterological Data Appendix E Circumsolar Data Index
491
423
452 453
483
440
Active Solar Collectors and Their Applications
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1. SURVEY OF SOLAR EQUIPMENT AND APPLICATIONS
1.1 THE PROMISE AND THE CHALLENGE OF SOLAR ENERGY Solar radiation is incident on the earth's surface at a rate of approximately 8 X 10'6 W, more than 10,000 times the present world energy consumption. Obviously the resource base is there to support a solar-energy-based civilization at the current level of energy use. In order to be useful, however, the energy must be collected and brought to the right place at the right time. A more meaningful number is therefore the flux density or radiation per unit surface area. The annual average solar flux on a horizontal surface in the U.S. is in the range of 150-200 W/m2.1 A typical American house has a roof area on the order of 100 m2; hence the solar flux incident on a roof is in the range of 15-20 kW. This is larger than the annual average energy consumption rate of a typical residence, which is on the order of 5 kW. This example indicates that it is not unreasonable to expect solar energy to make a very large contribution to our future energy supply. How much solar energy really will contribute depends on factors that are difficult to foresee at the present time. The acceptance of solar energy will depend on the cost of the equipment needed to collect it. Even though sunlight is free, one must pay a capital charge for the amortization of the collecting equipment. Solar energy is capital intensive. For solar equipment one usually assumes a lifetime on the order of 20 years and a useful rule of thumb is that the annual capital charge in constant dollars is about 10% of the initial investment.2 For example, if the equipment is financed by a 20 year loan at 8% interest (real, i.e., above inflation), then the annual payments for principal and interest are 10.2% of the initial investment. In addition, there will be some expenses for operation 'The global average insolation may be estimated as follows. Outside the atmosphere a surface normal to the sun receives 1373 W/m2; the so-called solar constant. Averaged over the surface of the earth this value is reduced by a factor of 4 because the cross section the earth presents to the sun is one-fourth of the earth's surface area. Passing through the atmosphere, the solar radiation is attenuated by approximately a factor of 2. Thus the average insolation at the earth's surface is (24 h average) 1373 W/m 2 X Vt X 'A = 170 W/m2. 2 The economics of solar investments is discussed in Chapter 14, and an explicit formula for the annual charge rate is given by Eq. (14.4.9).
3
4
Active Solar Collectors and Their Applications
and maintenance. These are usually small, on the order of 1 %-2% of the initial equipment cost, and for the moment we assume them to be included in the "10% rule." An example will illustrate the effect of capital charges. Consider a solar energy system at a location where the average insolation is 170 W/m2. The annual total solar radiation is (1 year =* 3.15 X 107 sec) approximately 5.4 GJ/m2. If the system has a 40% conversion efficiency into useful energy, then the useful energy is 2.1 GJ per m2 of the collector per year. Suppose this system costs $150 per m 2 of collector. According to the "10% rule," the annual cost is $ 15 per m2 of collector. This means that the cost of solar energy is
This is approximately the price of heating oil in 1980; hence this system is competitive with heating oil. Conversely, one can use the price of alternative energy sources to determine the highest permissible cost of solar equipment. For example, if the solar equipment is used for domestic hot water and if the house would otherwise burn oil at $7.1/GJ, then the value of the solar energy is
The purchase of solar collectors can thus be considered as an investment that returns an annual profit of $ 15 per m2 of collector. The above rule of thumb relates this annual return to the cost of capital and to the lifetime of the equipment. In this case the 10% rule says that one can afford to pay
for this particular system at this location. If insolation or efficiency are higher, one can pay proportionally more. This is a rather crude analysis and neglects many additional factors that should be considered; e.g., degradation of system performance with time, escalation of fossil fuel costs, and efficiency of oil furnace. As for general inflation, it affects annual charge rate and fuel costs equally, and it cancels out in the comparison; hence it is permissible to do the analysis in constant dollars. For more detail the reader is referred to Chapter 14. Above all we should, as a rational society, evaluate solar energy not in terms of market prices but in terms of full social costs. In particular, we should keep in mind that the market prices of fossil fuel and electricity fail to account for shadow costs due to pollution, radioactive wastes, oil supply disruptions, etc.
Survey of Solar Equipment and Applications 1.2
5
SURVEY OF APPLICATIONS
Our energy use3 includes several forms of delivered energy, in particular, heating and cooling, lighting, mechanical motion, and the energy needed to make the products we use or consume. In most cases the useful energy is different from the energy carrier or intermediate energy form that serves to transmit the energy to the point of use. In industrialized societies electricity is the universal favorite energy carrier because it is clean and versatile. However, it is difficult to store and it is expensive; typically its price is about 3 times as high as oil, reflecting the conversion efficiency. Oil and natural gas are also very desirable as energy carriers because they are clean and fairly versatile. Oil and other liquid fuels offer the great advantage of high volumetric energy density; hence they are easy to store. In fact for most transportation it is difficult to find a good alternative to liquid fuels. Coal has high energy density, but as a dirty fuel it is much more difficult to convert than oil or gas. Nuclear fuel has the highest energy density and it is the most difficult to convert. When considering renewable energy sources one should pay attention to the best possible match between available energy type and desired end use energy type. One could, for example, burn biomass to supply industrial process heat. However, as Table 1.2.1 shows, the conversion efficiency from solar energy to biomass4 is very low, on the order of only 1%; this is far les than the conversion efficiency of solar thermal collectors, around 30%-60°/o. Conversion efficiency plays a crucial role. At very low conversion efficiency land availability can become critical. For example, at a conversion efficiency of 0.2% the entire land area of the contiguous U.S. would be needed to supply the current U.S. energy demand. The conversion efficiency from solar energy to electricity around 10%-20% is also much lower than the efficiency of solar heat collection. These facts suggest that one should use solar thermal collectors when heat is the desired end use energy form. Biomass seems to be better suited as a source of liquid fuel and as a form of energy storage. To get an idea of how solar energy might best be used by a solar-energybased civilization, let us look at the current energy consumption pattern of the U.S. in Table 1.2.2. This pattern varies somewhat from country to country, and it may change with time as a result of technological developments. For instance, the heating and cooling loads of buildings are expected to decline with the implementation of energy conservation measures. The further one looks into the future the more difficult it is to predict the energy consumption and to foresee the most appropriate mix of energy supply technologies. Anyway, it seems reasonable to take Table 1.2.2 as a guide for suitable applications of solar energy in an industrialized country in the foreseeable future. 3 For a good discussion of our current and future energy use the reader is referred to Ross and Williams [1981]. 4 The conversion from biomass to high-grade fuels incurs additional losses. The efficiency of this process can be improved by using high-temperature point-focus solar collectors to supply energy for flash pyrolysis of biomass [Antal, 1978].
6
Active Solar Collectors and Their Applications
TABLE 1.2.1 Solar Energy Conversion Methods and Typical Annual Average System Efficiencies
System and collector type (i)
Heat (a) Low temperature (hot water, space heating, industrial process heat) Flat plates Evacuated tube Parabolic trough (b) High temperature (industrial process heat) Parabolic dish Central receiver
(ii) Electricity (a) Photovoltaic Flat plate Concentrating
(b) High temperature Central reciever Parabolic dish Point-focus Fresnel lens (iii) Biomass
Percent efficiency (relative to radiation incident on aperture)
Comments
30-60
For a given system the efficiency decreases with temperature, Hl her 8 temperature collectors tend to cost more -
5-15 15-25
Probably 10% is the minimum needed for commercial success. Photovoltaic systems could be deployed in small modules (e.g., on roofs).
10-25
Short term energy storage as heat.
0.1-2
Energy is stored as chemical.
Hot water in the residential and commercial sectors represents a small but significant fraction of our energy use (3.4%) and it is a very good match for solar thermal collectors. The demand for space heating and cooling is large but it is not clear yet how much of it can economically be met by solar collectors. Another very large consumer of heat is industry; approximately 16% of our total energy is consumed as heat for industrial processes ranging all the way from the sterilization of milk to the reduction of iron ore. To assess the suitability for solar energy, one has to look at the temperature requirements of the individual processes. The lower the temperature the better the match with solar energy. The temperature distribution of process heat is included in Table 1.2.2. It is not clear to what extent this table represents the true end use temperature of industrial process heat. In practice, most of the heat below 300°C is supplied by process steam. Most factories use a single steam system to cover a range of processes, and its temperature is above the highest temperature of all the individual processes. Steam is the most convenient heat transfer
Survey of Solar Equipment and Applications
1
TABLE 1.2.2 Energy Consumption of U.S. in 1977 by End-use, in Units of 1018 J/yr (Numbers in parentheses show percentage of total. Based on USDOC [1979] and SERI [1981]
C Transportation 21.2(26.2%) 8.r7 (10.8%) Space heat Residential 18.2
Residential plus commercial 29.9 (37.0%) Total 80.8 (100%)^
(U.i *>)
2.5 (3.1 %) Hot water 1.2 (1.5%) Air conditioning 2.4 (3.0%) Rdfrigerators and freezerzs u
(L40/()) Lights
I 2.3 (2.8%) Other Commercial 11.7 (14.5%)
f 5.2 (6.4%) 2.5 (3.1%) \ 0.26(0.3%) I 2.5 (3.1%) ^ 1.2 (1.5%)
Space heat Air conditioning Hot water Lights Other
r 1.0 (1.2%) Below 100°C Process neat heat process 12.0
Industrial
(16.0%)
29.7
(36.8%)
3.8 (4.7%) 100-177C 2.9 (3.6%) 177-288C 2.6(3.2%) 288-593 C 1.0 (1.2%) 593-1090C 1.6 (2.0%) above 1090 C
Other industrial 16.8 (20.8%)
fluid, and steam frequently is used even for heating water baths. Many factories with a central steam system have in fact, some processes at low temperatures; in a solar retrofit, one could disconnect these processes from the central system and provide their heat with low-temperature collectors. Thus the true distribution of end use temperatures may be shifted towards significantly lower temperatures than indicated by Table 1.2.2. However, for the sake of illustration let us proceed with the numbers in this table. A little over 1% of our total energy is used by industrial processes below the boiling point of water, suitable for simple nontracking collectors. Almost 5% of our total energy is process heat in the range of 100-177°C; this could be supplied by nontracking evacuated collectors. Parabolic trough collectors are available today that can operate efficiently up to 300°C. The total process heat below 300°C accounts for about 10% of our entire energy consumption. At higher temperatures the problems with solar collectors increase, and it is not yet clear how much of the very-high-temperature process heat could be met by solar heat. Electricity can be generated by solar thermal or by photovoltaic conversion. In the solar thermal approach the Carnot efficiency places a premium on high collector temperatures, and in practice one will probably choose point focus collectors. Low-temperature conversion is practical only if the source of heat is extremely cheap; this appears to be the case for ocean ther-
8
Active Solar Collectors and Their Applications
mal gradient power plants and for salt gradient solar ponds in favorable locations. The efficiency of high-temperature solar thermal power plants is comparable to that of photovoltaic systems. Photovoltaic systems appear to be simpler; they are modular and can be deployed wherever there are flat unshaded surfaces (e.g., on roofs). On the other hand, it is probably easier to include short-term storage in solar thermal plants. Storage of a few hours is very desirable in order to match solar electricity to the daytime demand peak experienced by most utilities. These numbers suggest a very large potential for the applications of active solar collectors. However, one also needs to consider the relation between demand and solar availability. The useful energy delivered by a solar energy system depends on the extent to which there is a load when solar energy is available. Most loads are not perfectly matched to solar availability, thus necessitating some kind of energy storage.5 The utilization of the solar equipment is best if there is a year-round load. This is the case for domestic hot water and for most applications in power generation and industrial process heat. For active solar space heating, by contrast, the collectors are idle during the warm part of the year and the permissible collector cost is reduced accordingly. Implicit in these statements is the assumption that these systems contain only short-term thermal storage. To improve the collector utilization of space heating systems, one can either add heat-driven chillers or seasonal heat storage [Atterkvist, 1979; Sillman, 1981]. Driving an air conditioner with heat from a flat plate collector is difficult and the cost effectiveness of this approach is dubious [SERI, 1980]. In fact, for solar cooling, the best approach may well be an electrical air conditioner with electricity from photocells or a central solar power plant (see Chapter 12 for more details). Finally, let us emphasize the role of cogeneration as a means of improving the efficiency of solar energy utilization. When converting solar energy to electricity only, one wastes most of the energy as heat. If instead one utilizes this heat to supply thermal needs, the overall efficiency of the system is greatly enhanced. This simultaneous production of several useful energy forms (e.g., heat and electricity) is called cogeneration. It is appropriate whenever thermal and electric energy demands are sufficiently well matched. Opportunities for solar cogeneration of heat and electricity abound in industries and in properly planned communities. Both photovoltaic and solar thermal approaches are practical for cogeneration. 1.3 SOLAR COLLECTOR TECHNOLOGY The most important and most expensive single component of an active solar energy system is the collector field. Therefore, we begin with a survey of solar collectors, summarized in Tables 1.3.1-1.3.3. The tables list collector types, typical operating temperatures, current costs, and additional cornThere is one load that is almost perfectly matched to the sun: the pumping power for active solar systems. This is not negligible, costing typically 1%-10% of the collected energy, and it may be an excellent application for photovoltaics.
9
Survey of Solar Equipment and Applications TABLE 1.3.1 Nontracking Collectors
Collector type
Approximate maximum operating temperature (°C)
Cost ($/m2) a
Comments
Shallow solar pond
40-60
160 (complete system, including storage for 1 day)
Plastic covers may need to be replaced every 5 yr or so. Needs sunny climate for good performance.
Deep solar pond (salt gradient)
40-90
30-60b (includes storage)
Collector and longterm storage in one unit. For seasonal storage, depth should be about 3 m. Low cost, but low efficiency (10%-20%).
40-80
150-300c'd
30-60 10-20 above ambient
70-lOOc'd 70-100c
Best known and most developed of all collector types.
Nonevacuated CPC fixed-tilt or summerto-winter tilt adjustment
80-120
150c-d
Evacuated tubes (with reflector enhancement; e.g., CPC)
100-200
250-300°'"
Flat plate (a) Conventional design (b) Made of plastic (c) Unglazed
Many opportunities for cost reduction by mass production and for performance improvements through R&D.
"Cost estimate in 1980. Personal communication, Solar Energy Group, Lawrence Livermore Laboratory, Livermore, CA 94550. b Costs of solar ponds are very site-specific (e.g., salt may be free, liner may not be needed, etc.). A 3-m deep salt-gradient solar pond was built in Miamisburg, OH for 35 $/m2 in 1978. c Solar Products Specifications Guide. 1983. Published annually by Solar Age Magazine, Church Hill, Harrisville, NH 03450. d For some collectors several manufacturers are in the market, with a wide spread in quality and price (not always correlated). Some models exceed the price range indicated here.
ments. With any thermal collector, the efficiency decreases as the operating temperature increases. The effects of operating conditions on the efficiency 17 of a solar collector can be described by the equation
where
and
170 = optical efficiency U = heat loss coefficient [in W/m 2 K] AT = difference between collector temperature and ambient temperature / = insolation on aperture [W/m2]
10
Active Solar Collectors and Their Applications
TABLE 1.3.2 One-Axis Tracking Collectors
Collector type
Approximate maximum operating temperature (°C)
Cost ($/m2)
Comments
Inflated cylindrical reflector
140
50-70"
Does not need continuous tracking, but does require weekly tilt adjustments; plastic cover may need to be replaced every 5 yr or so.
Parabolic trough
300
150-300b'c
Continuous accurate tracking; sensitive to dirt.
Line-focus Fresnel reflector
250
—
May combine advantages of parabolic trough and of central receiver for temperatures below 250°C.
Fixed line-focus reflector with tracking receiver
250
—
Problems with dirt accumulation on reflectors.
a Cost estimate in 1980. Personal communication, Solar Energy Group, Lawrence Livermore Laboratory, Livermore, CA 94550. b 'Solar Products Specifications Guide. 1983. Published annually by Solar Age Magazine, Church Hill, Harrisville, NH 03450. c For some collectors several manufacturers are in the market, with a wide spread in quality and price (not always correlated). Some models exceed the price range indicated here.
TABLE 1.3.3 Two-Axis Tracking Collectors
Collector type
Approximate maximum operating temperature (°C)
Parabolic dish or pointfocus Fresnel lens
1500 (possibly more)
Central receivers
1000 (possibly more)
Cost ($/ m2) —
492"
Comments Good if energy can be used directly in focal zone (e.g., photovoltaics or solar thermal power); otherwise, transporting heat to point of use is problematic. Optical transport of energy.
355b plus tower0 Fixed-hemispherical reflector, tracking receiver
400
Problems with heat transport to point of use, and with dirt accumulation on reflector.
"Average cost of heliostats for Barstow solar power plant, in 1980 [Battleson, 1981]. b lncrcmental cost of heliostats for Barstow after tooling costs, etc. have been paid, in 1980 [Battleson, 1981]. °Cost of tower is estimated to be approximately 10% of heliostat cost.
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The collector parameters 770 and [/are, to a first approximation, independen of operating conditions. There is no sharp cutoff for the typical operating temperatures; rather, the indicated temperatures are rules of thumb, based on current technology. High-temperature collectors could be run at lower temperatures than indicated in these tables, but that is not practical since low-temperature collectors are available at lower cost. 1.3.1
Flat plates and solar ponds
Flat plates and solar ponds, Figs. 1.3.1-1.3.3, are the most economical choices for low temperatures. However, because of high heat loss, they are quite sensitive to ambient temperature and insolation. In an economic comparison with other collector types, consideration of climate and operating temperature is therefore crucial. For swimming pool heating the temperatures are so low that unglazed collectors may be the most cost effective. Flat plates are the best-developed collectors and cost reductions are difficult to achieve, at least with current designs, which use mostly copper absorbers and tubes. Recently a promising low-cost lightweight plastic collector has been developed [Andrews and Wilhelm, 1980] for applications where the water quality inside the collector is not as critical as for domestic hot water; e.g., industrial process heat. Also, many American residential systems use double loops with heat exchanger and with antifreeze in the collector loop for frost protection; in such cases there is no need for copper absorber tubes. Among solar ponds, one must distinguish two entirely different types: the shallow solar pond and the deep (or salt-gradient) solar pond. The shallow solar pond, Fig. 1.3.2, consists of a shallow horizontal water bag, insulated by one or more plastic films and air layers [Clark and Dickinson, 1980]. It is filled in the morning and drained into a storage tank in the evening. Because of ultraviolet degradation, the outer cover of a shallow solar pond may need replacement, perhaps every 5 years. The deep solar pond, Fig. 1.3.3, uses a thick layer (about 1 m) of nonconvecting water as insulation [Tabor and Weinberger, 1980; Nielsen, 1980]. Convection is prevented by adding salt in such a way as to establish a concentration gradient, with the saltier water at the bottom. The saltier water is heavy enough to stay at the bottom even when warmed by the sun. A layer of 1 m of nonconvecting water offers as much thermal resistance as 5 cm of
Figure 1.3.1 Typical liquidheating flat plate collector (From Lof[ 1980]).
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Active Solar Collectors and Their Applications
Figure 1.3.2 Cross section of a shallow solar pond module (From Clark and Dickinson [1980]).
styrofoam, but it transmits much of the incident solar radiation. The concentration gradient can be maintained quite easily by washing the surface with fresh water while reinjecting concentrated brine at the bottom. The annual salt consumption is on the order of 10% of the salt inventory of the pond. Beneath the nonconvecting layer, there is a convecting layer of salt water for thermal storage and heat extraction; its thickness is in the range of 0.2-2 m, depending on the desired amount of storage. Figure 1.3.3 also indicates a convective zone of approximately 0.10-0.4 m at the surface. Even
Figure 1.3.3 (a) Cross section of solar pond, schematic, showing three-zone configuration, (b) Salinity profile, a possible stationary configuration, (c, d) Temperature profiles, idealized, anticipated in space-heating applications (From Nielsen [1980]).
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though undesirable, such a convective layer at the surface seems to be very difficult to avoid in practice. Both deep and shallow solar ponds perform well in sunny and southerly locations. In cloudier climates the achievable operating temperature may be too low to be of interest to industry; however, for space heating the deep solar pond may be attractive for apartment complexes and district heating systems [Rabl and Nielsen, 1975]. Deep solar ponds combine collector and long-term storage into a single low-cqst element and are currently the only suitable candidate for stand-alone solar installations. The salt contributes a large fraction of the pond cost and may create an environmental hazard in case of a leak. Therefore, salt-gradient solar ponds are best suited for areas where salt is cheap and where salt pollution would not be critical (e.g., the Dead Sea, the Great Salt Lake, and coastal areas). 1.3.2
Evacuated tubes
Because of their high heat loss coefficient, ordinary flat plate collectors are not practical for elevated temperatures, say above 80°C. When higher temperatures are desired, one needs to reduce the heat loss coefficient. This can be accomplished principally by two methods: evacuation and concentration, either singly or in combination. While several attempts have been made to build evacuated flat plates, they do not seem to hold any promise of commercial success. The natural configuration for an evacuated collector is the glass tube. There are many possible designs, and quite a few manufacturers are offering evacuated tubular collectors for sale. All of them use selective coatings as absorber because with a nonselective absorber, radiation losses would dominate at high temperatures, and eliminating convection alone would not be very effective. Figure 1.3.4 shows the most common evacuated collector types. The basic problem for the design of these collectors arises from the thermal expansion of the absorber relative to the glass tube. In the flow-through design of Fig. 1.3.4a a bellows is required to accommodate expansion. This necessitates glass-to-metal seals since bellows are made of metal. The design of Fig. 1.3.4b avoids the need for bellows by putting inlet and outlet at the same end of the tube. Figure 1.3.4b shows both legs of the hairpin fluid conduit attached to the absorber plate. That imposes a performance penalty, because it allows thermal short circuiting between the hotter outgoing fluid and the cooler incoming one. It would be more efficient to have one pipe attached to the center of the plate while the other pipe floats free in the space below the plate. A heatpipe6 provides the most elegant way of extracting heat from an 6 A heatpipe is a hermetically sealed tube that contains a small amount of heat transfer liquid. When one portion of the tube is heated the liquid evaporates and condenses at the cold portion, transferring heat with great effectiveness because of the latent heat of condensation. The heatpipe contains a wick or is tilted (or both) to ensure that the liquid flows back to the heated portion to repeat the cycle. It is easy to design a heatpipe (e.g., by giving it the proper tilt) so
Figure 1.3.4 (a) Sanyo Evacuated Tube Collector (From Graham [1979]). (b) Corning Cortec Evacuated Tube Collector (From Graham [1979]). (c) Philco Italiana Collector
with heatpipe (From Graham [1979]). (d) Glass dewar. (e) Owens-Illinois SUNPAK Evacuated Tube Collector (From Graham [1979]). (f) GE Collector (From Graham [1979]).
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Active Solar Collectors and Their Applications
evacuated collector. The collector in Fig. 1.3.4c has the same absorber plate configuration as Fig. 1.3.4b, but the heat is extracted by a heatpipe. The condensor end is relatively small (5-10-cm long compared to the length of the collector tube, 1-2 m), and it is inserted into a manifold that collects the heat from the individual tubes. The need for glass-to-metal seals can be avoided by the Dewar or thermosbottle design. Here the absorber tube is made of glass, sealed to the outer glazing tube at one end; at the other end, both tubes are closed, as shown in Fig. 1.3.4d. The selective absorber coating is applied to the outside (vacuum side) of the inner tube. There are several methods of extracting the heat from the absorber surface. The collector in Fig. 1.3.4e places a feeder tube concentrically inside the absorber tube. Heat transfer fluid enters through the feeder tube. The feeder tube is open at the other end and allows the fluid to return through the annulus between feeder and absorber tube. The performance penalty due to the thermal short circuit between incoming and outgoing fluid seems to be acceptable in this case. In the collector of Fig. 1.3.4f a copper sheet is placed inside the absorber touching the glass to obtain good heat transfer from glass to copper sheet. A hairpin fluid duct is bonded to the sheet. One can also use a heatpipe instead of a hairpin duct. While the thermal contact between sheet and glass does not appear to be good, it is good enough in practice because the thermal resistance from fluid to absorber is small compared to the resistance from absorber to ambient (see Section 10.1 for details). For the same reason the design of Fig. 1.3.4e can operate efficiently with air as the heat transfer fluid, despite the seemingly poor heat transfer from glass to air. Evacuated tubular collectors are hermetically sealed and contain getters to absorb any molecules that outgas into the vacuum. The tubes are expected to have a maintenance free lifetime on the order of 20 years; not an unreasonable expectation based on experience with vacuum tubes for radio and television. Evacuated tubes are excellent for operating temperatures up to the 120-150°C range. They are nontracking. Many of them use some kind of reflector enhancement. Evacuated tubular collectors have great potential for cost reduction through mass production, but the investment required to build efficient production facilities is too large to be justified by present demand. A photograph of a typical evacuated tubular collector is shown in Fig. 1.3.5. 1.3.3 Concentrating collectors Heat losses are approximately proportional to absorber area. By concentrating the radiation incident on the aperture onto a smaller absorber, one can reduce the heat loss per collector aperture area. Optical concentration that it functions only in one direction. This thermal diode effect is very useful for the design of solar collectors, because it automatically shuts the collector off and prevents heat loss when there is insufficient solar radiation. Also, heatpipes have lower heat capacity than ordinary liquid-filled absorber tubes, thus minimizing warm-up and cool-down losses.
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Figure 1.3.5 Evacuated tubular collector (Courtesy of Sunmaster, Inc. [1983]).
must inevitably reduce the field of view of the collector. Thus the higher the concentration, the lower the acceptance for diffuse insolation. Also it becomes necessary to follow the motion of the sun. This tracking requirement increases with concentration. Very low concentrations, generally in the range of 1-1.3 averaged over the year, can be reached by planar reflector enhancement. For concentration ratios up to 2, one can avoid the need for tracking by using the so-called CPC (compound parabolic concentrator) design, also called nonimaging concentrator. The CPC is well suited for evacuated tubes, and most of the evacuated collectors sold in the U.S. use CPC reflectors. Figure 1.3.6a shows the cross section of an evacuated tube with external cusplike CPC reflector. This is in fact the arrangement used
18
Active Solar Collectors and Their Applications
Figure 1.3.6 (a) CPC reflector coupled to tubular evacuated receiver, (b) Evacuated glass tube shaped as CPC reflector (From O'Gallagher et al. [1982]).
by the collector in Fig. 1.3.5. A more advanced design is shown in Fig. 1.3.6b, where the evacuated tube itself is shaped as a CPC reflector. Since the reflector is protected from the environment, a high-reflectance silver coating can be used. Recently, O'Gallagher et al. [1982] have achieved efficiencies of 50% around 20CTC with a fixed-tilt-evacuated collector using glass tubes shaped as CPC reflector, a design sketched in Fig. 1.3.6b. Thus the important market for industrial process steam at 175°C could be supplied by collectors that are quite insensitive to dirt and avoid the problems of tracking. In addition, they accept most of the diffuse insolation; they could therefore outperform tracking collecotors, especially in cloudier regions. CPC collectors with low concentrations and nonevacuated receivers may be economical for temperatures around the boiling point of water [CollaresPereira and Sequeira, 1982]. While CPCs are most suitable for low concentration ratios (i.e., for fixed collectors and for collectors with seasonal tilt adjustments), focusing optics are preferable for the design of tracking collectors. Collectors with high concentration require accurate and continuous tracking and are quite sensitive
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to dirt accumulation, on both the reflector and the receiver. Regular cleaning, perhaps every 2 weeks, may be necessary in dirty environments. Presently the parabolic trough, Fig. 1.3.7, is the favored collector for the temperature range of 150-300°C, and there are several manufacturers. Another candidate for process steam is the linear Fresnel reflector, Fig. 1.3.8, in which individual tracking mirror slats focus solar radiation onto a common linear receiver. One such collector with slats of approximately 0.2 m width is in commercial production. An interesting alternative to a concentrator with tracking reflector is a fixed reflector with tracking receiver as shown in Fig. 1.3.9. The main problem posed by the fixed reflector is dirt accumulation on the mirror. A tracking reflector is turned upside down whenever the sun is not shining, while a fixed reflector collects dirt at all times. Dirt accumulation is particularly severe in the early morning hours when dew condenses on dust particles in the atmosphere and then settles on exposed reflector surfaces. Furthermore, the reflector for the fixed reflector with tracking receiver lacks the proper tilt to be cleaned by rain. Collectors with line-focus Fresnel lenses, Fig. 1.3.10, pose a problem for large thermal installations. The difficulty stems from off-axis optical aber-
Figure 1.3.7 Parabolic trough (Courtesy of Sandia Laboratories [1978]).
Figure 1.3.8
Line-focus Fresnel reflector (Courtesy of Sandia Laboratories [1978]).
Figure 1.3.9 Line-focus reflector with tracking receiver (Courtesy of G. Eggers [1983]).
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Figure 1.3.10 (a) Ordinary lens, (b) Fresnel lens.
rations, which necessitate tracking either about the polar axis (north-south axis tilted at the latitude) or even worse about two axes, increasing the costs of collector mounting and plumbing. However, as concentrators for photovoltaic cells linear Fresnel lenses are well suited. Line-focus collectors have been tested to 350°C. Advanced components (evacuated receiver, improved coatings, high optical quality) could perhaps allow operation to 500°C. This has not been demonstrated and may be too expensive even if successful, not to mention the problem of transporting high-temperature heat from a distributed collector field to the point of use. Attainment of temperatures above 500°C (and probably above 300°C) requires point-focus concentrators. The principal types are the central receiver (also known as the power tower), the parabolic dish, and the point-
22
Active Solar Collectors and Their Applications
focus Fresnel lens. A photo of the first central receiver solar power plant in the U.S. is shown in Fig. 1.3.11 [see also ASME, 1984]. Point-focus parabolas and Fresnel lenses are good collectors for electric power if the generators are at each focus, but they suffer a severe drawback as far as process heat is concerned. It is very costly to collect high-temperature heat from a field of parabolic dishes or Fresnel lenses, and the pumping power and thermal losses are large. This effect is likely to outweigh the greater collection potential of the dish. The problem is less severe in an array of parabolic troughs because the absorber doubles as transport pipe. The central receiver avoids this problem by transporting the energy optically rather than thermally. Difficulties in transporting hot fluids increase with temperature and, for high temperatures the central receiver appears to be the only reasonable choice for large installations. Certain thermochemical reactions may permit energy transport at low temperature, but such schemes are not yet near commercialization. The alternative of transporting the products to be processed to the focal points of parabolic dishes is probably not compatible with efficient process operation in most industries. A possible process heat application of parabolic dishes might be found in installations that are too small for a central receiver.
Figure 1.3.11 Central receiver power plant at Barstow, CA (Courtesy of Sandia Laboratories [1983]).
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Survey of Solar Equipment and Applications
One last concentrator type should be mentioned: the hemispherical fixed reflector with tracking receiver. Like other fixed reflectors it is quite sensitive to dirt accumulation and, like the parabolic dish, it faces serious problems of heat transport from absorber to point of use. 1.4 THERMAL STORAGE TECHNOLOGIES
The most important thermal storage technologies for solar energy are listed in Table 1.4.1. With air systems, rock beds are the best known and most cost effective of storage techniques. For liquid collector loops, water seems to be the best storage medium below 100°C. Low-temperature phase change materials tend to be more costly and find their main justification in cases where space is at a premium; e.g., in space heating applications. Above 100°C, a water tank would have to be pressurized, and the cost increases rapidly with temperature. Oil is a likely alternative to water for high-temperature storage. Since oil by itself is quite expensive, a mixture of oil with TABLE 1.4.1 Storage Technologies Storage medium
Temperature range (°C)
Cost (in $/MJ) ab
Water
<100
0.5-2 '
Pressurized water
—
—
Water plus antifreeze Rocks Oil with rocks, iron, iron ore, or waste glass
100-120
— 0.5-2b 3-6c'd
120-300
Phase-change materials Thermochemical reactions
Depends on materials
Salt Liquid metal (sodium)
150-550 125-760
Comments As process temperature approaches 100°C, temperature range decreases and cost increases. Cost increases rapidly with temperature. — Excellent with air systems. Due to heat exchangers, pumps, etc., actual cost is much higher (about twice as high for 6-h storage). Permits long term storage at ambient temperature. Chemical heat pump may yield efficiencies greater than unity.
"Cost in $/MJ depends on temperature swing in addition to costs of material and container. Water tanks in the range of 1000-10,000 liters cost about $ 0.2-0.5/liter [Solar Product Specifications Guide, 1983]. b Baylin, F. SERI. Personal communication [1980]. G Copeland, R. J. et al. "A Preliminary Screening of Thermal Storage Concepts for Water/Steam and Organic Fluid Solar Thermal Receiver Systems." "Office of Technology Assessment. 1978. "Application of Solar Technology to Today's Energy Needs." Volume 1.
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Active Solar Collectors and Their Applications
rocks, waste glass, or other low-cost materials is often preferred. Such an arrangement also offers the benefit of improved thermal stratification. For high temperatures molten salt is a candidate. Of the other less-developed storage methods, thermochemical reactions, in particular the chemical heat pump, are promising because they permit long-term storage. Storage tanks have a tendency towards stratification, with warmer fluid rising above colder fluid. In solar thermal systems the fluid from the bottom of the tank is pumped to the collector and the heated fluid from the collector is returned to the top of the tank. Therefore, stratification improves collector efficiency by lowering the collector temperature relative to a well-mixed tank. Depending on the load, the benefits from stratification may be further enhanced. For example, in a domestic hot water system a stratified tank can deliver useful hot water earlier in the day than a fully mixed tank. If the fluid from the tank is used to drive a heat engine or other equipment whose efficiency increases with temperature, then a stratified tank results in higher equipment efficiency. In practice one does not always achieve stratification. Mixing is caused by conduction through the tank wall and through the fluid itself. Furthermore, the flow of the fluid through the system tends to stir the tank. Tanks that contain only a liquid (e.g., water or oil) are likely to be partially mixed. One could improve stratification by choosing tall and narrow tanks, but that is likely to be costly, to say nothing about the extra heat losses from the increased surface area. On the other hand, loosely packed rock beds are very good at preserving stratification. Rock beds for air systems and tanks with a mixture of oil and rocks are highly stratified. 1.5 STORAGE, BACKUP, AND THE ELECTRIC UTILITIES
A fundamental problem of solar energy lies in the intermittency of solar radiation. Even in a perfectly cloudless climate the sun does not shine at night. In most areas, especially the populated areas, sunlight is further reduced by clouds that appear in more or less irregular patterns.7 Since much of our energy demand does not coincide with the availability of solar radiation, some form of energy storage or backup is essential. Whether it is better to store solar energy or to use backup in the form of oil, gas, or electricity depends on many factors. The user of energy does not care how the energy is supplied as long as it is available when needed and at the lowest possible cost. Therefore, one must consider the entire energy supply system, not just the solar components. The most important factors to be considered are listed below: 1. type of energy demand (e.g., heat, cooling, electricity) 2. time of energy demand (e.g., continuous, part of day, part of year) 7 For these reasons, G. K. O'Neill, author of The High Frontier, has said that the earth's surface is one of the worst places for collecting solar energy. A satellite in outer space can be kept unshaded and normal to the sun at all times, and thus it can collect more than 5 times as much radiation as a collector on earth.
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3. type of backup (e.g., oil, gas, electricity) 4. type of storage (e.g., batteries for solar cells, hot water tank for solar heater) As for energy storage, a simple rule of thumb is that the cost of stored energy is inversely proportional to the number of times storage is charged and discharged (for a more thorough analysis, see Chapter 15). In practice, storage of solar heat appears to be practical for short periods, on the order of hours or at most a few days. Seasonal or annual storage of energy is economical only if the storage material is extremely cheap; water, salt, rock, and soil seem to be the only candidates for long-term heat storage. Storage of electricity is costly, and probably not practical for periods longer than a few hours. Solar energy systems without long term storage must have complete backup. For solar thermal installations oil is an excellent backup. Because of its high volumetric energy density it is easy to store, and as a clean fuel it can be burned easily. Coal is an awkward backup fuel for solar despite its high-energy density; it is a dirty fuel and requires expensive combustion and cleanup equipment. If electricity is used as backup, the utility company may encounter problems. If a utility has to supply backup electricity for a solar installation, it may have to build additional generating capacity to meet this demand, even though this extra capacity will be idle much of the time. As a result, the utility may have to demand a high charge for supplying backup service [Asbury and Mueller, 1977]. The difficulties of interfacing solar energy systems with an electric utility depend on the load profile and on the generating equipment of the utility. Nuclear power plants are very capital intensive while nuclear fuel costs little [see, for instance, Komanoff, 1981]; hence they are economical only as baseload plants. Coal power plants are also expensive and their fuel is cheap. Like nuclear plants coal power plants are most economical as baseload plants, but they can also be operated at somewhat lower capacity factor; i.e., for intermediate loads or shoulder periods. Oil- or gas-fired power plants, on the other hand, are fuel intensive while their equipment costs are low; hence they are a good complement for solar. Most utilities experience a daytime demand peak due to their commercial customers (e.g., office buildings, shopping centers). This corresponds fairly well to the availability of sunshine. A solar plant with a few hours of storage can be a good choice for a summer peaking utility because summer peaks are due to cooling demand, which is more or less correlated with insolation, with a time lag of a few hours. In some situations a utility may also be able to benefit from the storage capacity of solar systems by storing low-cost off-peak energy at the customer's site. Industrial energy demand seems to be well matched for solar, because most industries operate year round. Much of the industrial process heat is provided by oil or natural gas and the interface with solar collectors is excellent. Furthermore, almost all industries have at least some demand during daylight hours and they can use some solar energy without the cost and performance penalties of storage.
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Active Solar Collectors and Their Applications
REFERENCES ASME. 1984. The February 1984 issue of the ASME J. Solar Energy Engineering 106:22-103, is devoted to central receivers. Andrews, J. W. and Wilhelm, W. G. 1980. "Thin-Film Flat-Plate Solar Collectors for Low-Cost Manufacture and Installation." Report BNL 51124. Upton, NY: Brookhaven National Laboratory. Antal, M. J. 1978. "Tower Power: Producing Fuels From Solar Energy." In Chapter 10, Toward A Solar Civilization, Williams, R. H., editor. Cambridge, MA: MIT Press. Asbury, J. G. and Mueller, R. O. 1977. "Solar Energy and Electric Utilities: Should They Be Interfaced?" Science 195:445. Atterkvist, S. 1979. "Swedish Energy Storage Projects 1979." Report D19: 1980 ISBN 91 -540-3257-1. Stockholm, Sweden: Swedish Council for Building Research. Battleson, K. W. 1981. "Solar Power Tower Design Guide: Solar Thermal Central Receiver Power Systems. A Source of Electricity and/or Process Heat." Report DE-81-02-65-84 SAND 81-8005. Albuquerque, NM: Sandia National Laboratories. Clark, A. F. and Dickinson, W. C. 1980. "Shallow Solar Ponds." In Solar Energy Technology Handbook, Dickinson, W. C. and CheremisinofF, P. N., editors. New York: Marcel Dekker. Copeland, R. L. et al. 1980. "A Preliminary Screening of Thermal Storage Concepts for Water/Steam and Organic Fluid Solar Thermal Receiver Systems." SERI/TR631-647. Golden, CO: Solar Energy Research Institute. Graham, B. J. 1979. "A Survey and Evaluation of Current Design of Evacuated Collectors." Final Report Contract No. DE-AC04-78CS05350. Annapolis, MD: Trident Engineering Associates, Inc. Romanoff, C. 1981. "Power Plant Cost Escalation—Nuclear and Coal Capital Costs, Regulation and Economics." New York: Komanoff Energy Associates. Lof, G. O. G. 1980. "Flat Plate and Nonconcentrating Collectors." In Solar Energy Technology Handbook, Dickinson, W. C. and Cheremisinoff, P. N., editors. New York: Marcel Dekker. Nielsen, C. E. 1980. "Nonconvective Salt-Gradient Solar Ponds." In Solar Energy Technology Handbook, Dickinson, W. C. and CheremisinofF, P. N., editors. New York: Marcel Dekker. O'Gallagher, J. J., Snail, K., Winston, R., Peek, C., and Garrison, J. D. 1982. "A New Evacuated CPC Collector Tube." Solar Energy 29:575. Rabl, A. and Nielsen, C. E. 1975. "Solar Ponds for Space Heating." Solar Energy 17:1. Ross, M. H. and Williams, R. H. 1981. Our Energy: Regaining Control. New York: McGraw-Hill. SERI. 1980. "Solar Heating and Cooling Systems: Operational Results." In Conference Proceedings, November 27-30, 1979. SERI Report SERI/TP-245-430. SERI. 1981. A New Prosperity—Building A Sustainable Energy Future. Andover, MA: Brick House Publishing. Sillman, S. 1981. "Performance and Economics of Annual Storage Solar Heating Systems." Solar Energy 27:513. Solar Products Specification Guide. Published annually by Solar Age, Church Hill, Harrisville, NH. Sunworld. 1981. Issues 3 and 4 ofSunworld, Volume 5, are devoted to solar thermal power generation. The central receiver, in particular, is described in Issue 4.
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Tabor, H. and Weinberger, Z. 1980. "Nonconvecting Solar Ponds." In Solar Energy Handbook, Kreider, J. F. and Kreith, F., editors. New York: McGraw-Hill. U.S. Department of Commerce. 1979. Statistical Abstract of the United States. Washington DC: U.S. Department of Commerce, Bureau of the Census, Government Printing Office. Williams, R. H., editor. 1978. Toward A Solar Civilization. Cambridge, MA: MIT Press.
2. SOLAR GEOMETRY
He had bought a large map representing the sea, Without the least vestige of land: And the crew were much pleased when they found it to be A map they could all understand. Lewis Carroll
The radiation received by a solar collector depends on the incidence angle. We therefore develop in this chapter the geometric relationships that determine the incidence angle. In many instances it will be convenient to have a terminology that distinguishes between radiative power and radiated energy. We shall designate instantaneous power [in W/m2] by the term "irradiance" and total radiated energy during a time interval such as 1 day by the term "irradiation" [in J/ m2]. We shall use the vague word "insolation" only when distinctions are not necessary. 2.1 THE SOURCE OF RADIATION The sun is a sphere of diameter 1.39 X 109 m and its average distance from the earth is 1.495 X 10" m. The interior of the sun is extremely hot, with temperatures of many millions of degrees. The surface temperature is, however, much lower, approximately 6000 K. One can define the effective blackbody temperature of the sun as the temperature of a blackbody radiating the same amount of energy per unit surface area as the sun. The effective blackbody temperature of the sun is 5762 K. Other effective temperatures could also be denned; for example, the temperature of a blackbody with the same wavelength of maximum radiation as the sun is approximately 6300 K [Duffie and Beckman, 1980]. The radiation emitted by the sun is nearly constant. The intensity of this radiation can be characterized by the solar constant 70. The solar constant is denned as the solar irradiance at normal incidence just outside the earth's atmosphere when the sun-earth distance is at its mean value of 1.495 X 10" m. (Due to the eccentricity of the earth's orbit the radiation incident on the earth varies with season by ±3.3%.) Measurements of the solar constant have improved over the years, espe28
Solar Geometry
29
cially since the beginning of space flights. In 1971 the National Aeronautics and Space Administration [NASA, 1971] accepted the value 70 = 1353 W/ m2, and much analysis has been based on this number. Since then, slightly larger values have been recommended. Frohlich [1977] found 70 = 1377 W/ m2. The most recent measurements by Hickey et al. [ 1982] have obtained a mean value with a standard deviation of approximately 1 W/m2. Even though for terrestrial solar applications, one needs to know only the insolation at the earth's surface, the extraterrestrial insolation has been used as the basis for certain correlations for predicting terrestrial insolation. A particularly useful correlation parameter is the clearness index Kr, denned as the ratio of terrestrial and extraterrestrial insolation on a horizontal surface. In order to avoid errors in such correlations due to changes in 70, it is advisable to use consistently the value of 70 employed by the authors who derived a particular correlation or graph of this kind. In any case, the change from 1353 W/m 2 to 1373 W/m2 is almost an order of magnitude smaller than typical uncertainties of terrestrial insolation data. 2.2
SUN-EARTH GEOMETRY
The earth moves about the sun in an elliptical orbit, completing one revolution per year. In addition, the earth rotates once per day about its polar axis. The polar axis is inclined at an angle of 23.45° from the normal to the plane of the earth's orbit (ecliptic), as shown in Fig. 2.2.1. These rotations
Figure 2.2.1 (a) Motion of the earth about the sun. (b) Location of tropics. Note that the sun is so far from the earth that the rays of the sun reaching the earth are nearly parallel to one another (Adapted from Kreith and Kreider [1978]).
30
Active Solar Collectors and Their Applications
and the inclination between polar axis and ecliptic cause the day/night cycle and the seasons. To describe the diurnal and seasonal variations of the incidence angle of solar radiation at the earth's surface it is convenient to adopt a ptolemaic viewpoint; i.e., to use a coordinate system that is fixed in the earth and to assume that the sun moves in a circular orbit around the earth. This is an acceptable approximation for solar energy calculations because the eccentricity of the earth's orbit is very small. There are, however, two features of an elliptical orbit that require special consideration. First, the distance between sun and earth changes by ± 1.7% between apogee and perigee. This effect can be accounted for by assuming that the effective solar constant varies with time of year according to the formula
where n is the day of year after 1 January and 70 is the solar constant = 1 373 W/m2. (Because of peculiarities of the earth's orbit, the peak value of 70eff occurs 10 days after winter solstice.) Second, there is a difference between real time and solar time. For a given point on the earth's surface in the northern (southern) hemisphere, solar noon is denned as that time of day when the sun appears due south (north). Solar time is the time of day measured from solar noon. Solar time coincides with real time only at certain times of the year; e.g., when the earth is at the perigee or apogee of its orbit. At other times, real and solar time may differ by as much as ± 1 5 minutes. The difference between local solar time and local standard time (corrected for longitude) is called the equation of time E. The value of E can be determined from Fig. 2.2.2 or from the equation [Duffie and Beckman, 1980] E = 9.87 sin 2B - 7.53 cos B - 1.5 sin B
[in minutes]
(2.2.2)
with B = 360°(« - 81)/364 for the nth day of the year.1 In terms of E, the solar time is related to standard time by solar time = standard time + 4(Lst — Lloc) + E
[in minutes],
(2.2.3)
where Ls, is the standard meridian for the local time zone, and L!oc is the longitude of the location in question in degrees west.2 Having taken account of the effective solar constant and the equation of time, one can assume that on any single day the sun completes one circular 'Recently a more accurate (and more complicated) equation has been published by Lamm [1981]. 2 Standard meridians for continental U.S. time zones are Eastern, 75°W; Central, 90°W; Mountain, 105°W; and Pacific, 120°W. There may also be an additional 1-h correction for daylight saving time.
Solar Geometry
31
Figure 2.2.2 The equation of time, E, in minutes, as a function of time of year (From Duffie and Beckman [1980], with permission of John Wiley & Sons).
orbit around the earth. This motion is described in solar time, and henceforth throughout this book all time-of-day designations are given as solar time. The solar orbit makes an angle 5 with the plane of the earth's equator. This 5 is called declination and varies with day of year n (after 1 January) according to
To derive this equation, vector algebra is convenient. Let us introduce into Fig. 2.2.1 (x, y, z) coordinates that are fixed in the sun. The z axis is normal to the plane of the ecliptic and the x axis points toward the earth at winter solstice. Then the unit vector n,, from earth to sun is given by
with
32
Active Solar Collectors and Their Applications
The polar axis of the earth is normal to the equatorial plane and has the direction of the unit vector
During the course of one day the earth completes one rotation about its polar axis; viewed from the earth (ptolemaic picture) the sun completes one orbit around the earth. The declination d is denned as the angle between the sun and the equatorial plane. Hence the angle between ne and the polar axis is 90° — <5. Taking the dot product ofne and np one obtains
The explicit result Eq. (2.2.4) follows directly when the coordinate values are inserted.
2.3 INCIDENCE ANGLES ON FIXED SURFACES The incidence angle of solar radiation on various tracking or nontracking collector apertures is calculated most easily by means of vector algebra. Let (x, y, z) be a Cartesian coordinate system fixed in the earth, with the z axis along the polar axis, as shown in Fig. 2.3.1. The x and y axes are chosen relative to the longitude where the collector is located, the y axis pointing east, and the x axis pointing towards the sun at noon on equinox. In these coordinates the unit vector ns from the earth to sun is given by
where <5 = declination, and w = hour angle, related to solar time t by
where rday = 24 h = 86,400 sec = length of day. The incidence angle 6 of solar radiation on a surface with surface normal n is then obtained from the dot product
For example, the normal n/, to the horizontal surface at a geographic altitude A is
Solar Geometry
33
Figure 2.3.1 Coordinate system for calculating incidence angles, x axis points towards solar noon, y axis points east, z axis = axis of rotation (north is +), 8 = solar declination, /? = collector tilt from horizontal (up is +), X = latitude (north is +), /30 = collector tilt from equatorial plane = £ — X, nc = unit vector normal to collector, n,, = unit vector normal to earth's surface, ns = unit vector in direction of sun.
Hence, the incidence angle of the sun on a horizontal surface is
This angle, shown in Fig. 2.3.2, is frequently called the zenith angle. Sunset3 occurs when the zenith angle reaches 90°; the corresponding sunset hour angle us is given by
and the sunset time ts is
Figure 2.3.3 shows a convenient nomograph from which the sunset time can be read directly. In terms of
34
Active Solar Collectors and Their Applications
Figure 2.3.2 Angular coordinates for orientation of collector and of sun relative to horizontal surface (indicated by rectangular frame with four cardinal directions), n' is normal of collector aperture. Dotted lines show projections in horizontal plane. /3 = tilt; 4> = azimuth of collector. 6, — zenith angle;
To find the incidence angle of the sun on a fixed collector of arbitrary orientation, let us use tilt /J and azimuth 0 to specify the orientation of the collector relative to the earth's surface, as shown in Fig. 2.3.2. In terms of
35
Solar Geometry
Figure 2.3.3 Declination and sunset nomograph, with example (From Whillier [1965], as adapted by Duffie and Beckman [1980], with permission of John Wiley &
Sons).
the (xr, y, z') coordinate system of this figure the collector normal n'c has the components
since the y' axis points east and the z' axis, north. The primed coordinates of Fig. 2.3.2 are obtained by rotating the unprimed coordinates of Fig. 2.3.1 through an angle X about the y = y' = east-west axis; hence
Applied to ns of Equation (2.3.1), this implies that the unit vector n's to the sun has the following coordinates in the xf, /, z' system:
Now the incidence angle 8 of the sun on the collector follows directly from the dot product
36
Active Solar Collectors and Their Applications
Sometimes the azimuth fa of the sun is of interest. To obtain an explicit expression for fa as a function of time we write n's directly in terms of the angular variables of Fig. 2.3.2 [analogous to Eq. (2.3.8) for n'] as
Equality of the y coordinates in Eqs. (2.3.10) and (2.3.12) implies
As a simpler case, consider a flat plate collector facing due south and tilted at an angle j3 from the horizontal. It is convenient to specify the tilt with respect to the equatorial plane as
hence the incidence angle is
This can be written as
For the important case of tilt equal latitude, /J0 vanishes and one finds
At a time of day and year corresponding to hour angle o> and sunset hour angle o>, the extraterrestrial solar irradiance on a horizontal surface is
with the effective solar constant of Eq. (2.2.1). Integrating over time of day from sunrise to sunset, one obtains the daily extraterrestrial solar irradiation
Solar Geometry
37
on a horizontal surface
with Tday = 24 h = length of day. Numerical values for //0 as a function of latitude and time of year are listed in Table D. 1 of Appendix D. There is a considerable decrease of the daily irradiation on a horizontal surface with latitude during the winter— when indeed there are 24-h nights above the Arctic Circle—but it varies surprisingly little from equator to pole during the summer. This is, of course, the principal explanation for the fact that it is so cold in the northern regions in winter while the differences between polar and equatorial temperatures are much less pronounced during summer. 2.4 INCIDENCE ANGLES ON TRACKING SURFACES
For tracking collectors nc depends on the tracking mode. For the case of an aperture with full 2-axis tracking, the collector always points to the sun, nc = Aj, and the cosine of the incidence angle is unity at all times:
As a more complicated example, consider an aperture which tracks about a north-south axis inclined at a tilt /3 from the horizontal. As before, it is convenient to refer the tilt to the equatorial plane as
Let (xf, y, z') be the coordinate system obtained by rotating the (x, y, z) system of Fig. 2.3.1 about the y axis through an angle j30. The components of a vector (x, y, z) are transformed according to
T! is the tracking axis of the collector. Designating the tracking angle relative
38
Active Solar Collectors and Their Applications
to noon by ^NS, one can write the collector normal in primed coordinates as
To keep the collector focused, one requires that n£ point in the same direction as the projection of the solar vector n^ onto the xf y plane (plane normal to the tracking axis). In primed coordinates, this requirement can be stated as the equality of the ratio of the x' and y components of h'c and n's:
Combining Eqs. (2.3.1) and (2.4.3) one finds
Equations (2.4.4) and (2.4.6) are now inserted into Eq. (2.4.5) to yield the tracking angle ^>NS in the form
The incidence angle 0NS is obtained from the dot product of n'c and n's:
Inserting Eq. (2.4.7) for the tracking angle and using some trigonometric identities one arrives at the following result for the incidence angle:
An interesting special case is the so-called polar mount, A = ft, where the tracking axis is parallel to the polar axis of the earth. With ft, = 0, Eq. (2.4.9) reduces to the simple and obvious result
The cosine of the incidence angle for polar mount ranges from 0.92-1.0 with an annual average of 0.96:
Solar Geometry
39
Therefore 1-axis tracking about the polar axis approaches within 4% the radiation availability of a collector with full 2-axis tracking. As another example, consider a collector with horizontal east-west tracking axis. In the (x, y, z) coordinate system of Fig. 2.3.1 the collector normal is
where the tracking angle ^EW is measured from the equatorial plane. Focusing requires that fic point in the same direction as the projection of the solar vector n^ onto the x, z plane (equals plane normal to tracking axis). Thus the x and z components of ns and nc must satisfy the relation
Inserting Eqs. (2.3.1) and (2.4.12), one finds the tracking angle ^EW for an aperture that tracks about the horizontal east-west axis as
The angle of incidence is obtained from the dot product ns • nc and turns out to be
The incidence angles for the principal aperture types are summarized in Table 2.4.1.
2.5 PROJECTED INCIDENCE ANGLES Collectors with reflector troughs or with tubular receivers have different optical properties along different directions. For example, the tubular collector in Fig. 2.5.1 has an optical efficiency i?o(0y, #1) that depends on two orthogonal angular coordinates 6% and 0j_. The CPC trough in Fig. 2.5.2 has a full 180° acceptance angle (or field of view) in the 0, direction along the trough but a limited acceptance angle 28a in the 0± direction perpendicular to the trough.4 We call the angles 0, and 8L projected incidence angles; they are the angles between the collector normal and the projections of the incident ray onto the planes perpendicular and parallel to the trough or tube axis, as shown in Figs. 2.5.1 and 2.5.2. Projected incidence angles are also 4
The subject of CPC and acceptance angle is discussed in Chapter 6.
40 TABLE 2.4.1
Active Solar Collectors and Their Applications Cosine of Incidence Angle 9 for Principal Collector Orientations
Orientation of collector aperture
cos 8
Fixed aperture: azimuth 0 = 0 Tilt 0 = latitude X Tilt @ <£ latitude X Fixed aperture: azimuth <j> ¥= 0
Aperture tracking about horizontal east-west axis Aperture tracking about polar axis Aperture tracking about horizontal north-south axis Aperture tracking about vertical axis Aperture tracking about northsouth axis inclined at angle /3 from horizontal Aperture with full 2-axis tracking
1
useful for the analysis of shading problems; in architecture they are known as profile angles. In most cases, collectors of this type will be installed with trough or tube axis pointing along either the east-west or along the north-south direction. For these orientations we calculate the projected incidence angles explicitly. For simplicity, let us consider the important case where the collector normal lies in the equatorial plane of the (x, y, z) system of Fig. 2.3.1:
this corresponds to a collector mount with tilt equal latitude, facing due south. Then the relevant projection planes are the (x, y) and the (x, z) planes in the coordinate system of Figs. 2.5.1 and 2.5.2. With the trough axis eastwest as in Fig. 2.5.2, the angle 6± is the angle 6XZ between the x axis and the projection
of the solar incidence direction n^ of Eq. (2.3.1) on the (x, z) plane, obtained by setting the y component of n^ equal to zero. dxz is found from the dot
Solar Geometry
41
Figure 2.5.1 Projected incidence angles 0, = 6xy and 6± = Bx, for tubular collector. 6 is the angle of incidence on the collector plane.
product
The result
can be rewritten in the more convenient form
42
Active Solar Collectors and Their Applications
Figure 2.5.2 Projected incidence angles 0, = 0,,, and d± = 6X- for CPC with eastwest trough, mounted at tilt equal latitude (coordinates as in Fig. 2.5.1.). (For the example shown, 0j_ > 8a and ray is rejected.)
It is plotted in Fig. 2.5.3 as function of hour angle u and for 0, 30, 60, and 80 days from summer solstice. Only the first quadrant is shown because the values for negative u or <5 follow by symmetry. The absolute value of 16 XZ \ always reaches its minimum
at noon. In summer, 5 is positive and dxz is greater than <5; the sun is always above the equatorial plane. In winter, the sun is always below the equatorial plane, <5 is negative, and 6X, is less than 5. For an application, let us return to Fig. 2.5.2, where a CPC is mounted at tilt equal latitude. Since the trough is aligned in the east-west direction, 0L
Solar Geometry
43
Figure 2.5.3 Projected solar elevation 0 V . (measured from equatorial plane) 0, 30, 60, and 80 days from solstice, as labeled by the number next to the curves. (6X. is the solar incidence angle projected on plane spanned by polar axis and by sun at noon.) The dotted line corresponds to the acceptance half-angle Sa = 35.5°, which a stationary collector must have in order to collect direct sunlight for at least 7 h/day.
equals
of Eq. (2.5.5):
This angle determines whether solar radiation is accepted or not. If \$xz\ is smaller than the acceptance half-angle 9a the ray is accepted; otherwise it is rejected:
Inserting \8XZ\ = 6a into Eq. (2.5.5) one finds the explicit expression for the cutoff angle coc
the cutoff time tc is of course related to wc by
For CPCs with concentration greater than 2 the tilt ft will generally be different from the latitude X, with tilt adjustments during the course of the year In that case the cutoff angle is given by
44
Active Solar Collectors and Their Applications
For a CPC with tilt adjustments one should always verify that the sun at noon is within the acceptance angle and that | wc | < 90°. EXAMPLE 2.5.1
A CPC with acceptance half-angle 6a = 35.5° is mounted at tilt equal to latitude. How many hours per day can it collect direct solar radiation at solstice? SOLUTION
The cutoff time occurs when dxz of Eq. (2.5.5) equals the cutoff angle 6a. For da = 35.5° this is indicated by the dotted line in Fig. 2.5.3. It intersects the curve corresponding to solstice (labeled with 0) at tc = 3.5 h; hence the collector accepts direct solar radiation for 2tc = 7 h at solstice. At all other times of the year the collection time is longer, reaching up to 12 h at equinox. As an example of shading, consider a south facing window of height h with an awning of width w as shown in Fig. 2.5.4. Assume that the awning is long compared to the window so that end effects can be neglected. Let us find the times when the window is completely shaded. From the geometry of Fig. 2.5.4 we see that the edge of the shadow falls on the bottom of the window when the projected angle 6XZ satisfies
where Combining Eqs. (2.5.5) and (2.5.12), we find the following relation for the corresponding hour angle wshadc
or
Only solutions with |w shade | < 90° are relevant, because at other times a south facing vertical wall does not receive any direct sunlight anyway. If a > X, the window is completely shaded during all of spring and summer. A convenient graphical method of solving Eq. (2.5.15) involves the use of a sun-path diagram and a shadow angle protractor; this is described in Chapter 2 of Kreith and Kreider [1978]. Finally we consider the projected angle of incidence 6XV in the equatorial
Solar Geometry
45
Figure 2.5.4 Shading by window awning over south facing vertical wall. (Projection on xz plane.)
plane. It is obtained from the dot product
where
is the projection of the incidence direction ns of Eq. (2.3.1) onto the (x, y) plane. It is easy to see that Bxy equals the hour angle & at all times:
46
Active Solar Collectors and Their Applications
For an application, let us return to the tubular collector in Fig. 2.5.1. If this collector is mounted at tilt equal latitude, with tubes running north-south, then 0t = 8X. and 6 L = 6xy. The optical efficiency ??o(0|, 0±) determines how much of the incident solar radiation 7 is absorbed by the absorber. When the solar declination is 5 and the hour angle is u>, the absorbed radiation is therefore [from Eqs. (2.5.5) and (2.5.18)]
EXAMPLE 2.5.2
A tubular collector mounted due south at tilt equal latitude has an optical efficiency that can be described as a product of a transverse and a longitudinal incidence angle notifier:
At normal incidence the optical efficiency is ?;0(0,0) = 0.60. Find the optical efficiency at 4 pm solstice if K/fl,) is given by the "2 glass cover plates" curve of Fig. 4.3. la, and K L (#i), by the "sunpak drainable" curve of Fig. 4.3.2. SOLUTION
At 4 pm the hour angle is w = 60°, and at solstice the declination is 5 = 23.45°. Hence
From Fig. 4.3.la the "2 glass cover plates" curve yields
while the "sunpak drainable" curve of Fig. 4.3.2 yields
Hence the optical efficiency at this time is
REFERENCES Duffie, J. A. and Beckman, W. A. 1980. Solar Engineering of Thermal Processes. New York: John Wiley & Sons. Frohlich, C. 1977. "Contemporary Measures of the Solar Constant." In The Solar Output and its Variation, White, O. R., editor. Boulder, CO: Colorado Associated University Press.
Solar Geometry
47
Mickey, J. R. et al. 1982. "Extraterrestrial Solar Irradiance Variability: Two and Onehalf Years of Measurements from Nimbus 7." Solar Energy 28:443. Kreith, F. and Kreider, J. F. 1978. Solar Energy Engineering. New York: McGrawHill. Lamm, L. O. 1981. "A New Analytical Expression for the Equation of Time." Solar Energy 26:465. NASA. 1971. "Solar Electromagnetic Radiation." Report NASA SP-8005. National Aeronautics and Space Administration. Rabl, A. 1976. "Comparison of Solar'Concentrators." Solar Energy 18:93. Whillier, A. 1965. "Solar Radiation Graphs." Solar Energy 9:164.
3. INSOLATION DATA AND MODELS
Always keep a record of data—it indicates you have been working. modern folklore, from Murphy's Laws
3.1 SOLAR SPECTRUM During its passage through the atmosphere the radiation from the sun is affected by scattering and absorption due to air molecules, water vapor, ozone, carbon dioxide, clouds, and dust. Scattering changes the direction of radiation in a random fashion and is the origin of diffuse sky radiation. Absorption is due mostly to O3 (ozone) in the ultraviolet and due to water vapor in the near infrared. A convenient measure of the path length of solar radiation through the atmosphere is the air mass ratio m (which is often called simply air mass). This quantity m is related to the zenith angle Oz by'
Figure 3.1.1 shows several spectral distributions for solar radiation.2 The x axis is the wavelength in /^m and the y axis is the spectral intensity [W/ m2 X /^m] of direct solar radiation. The highest curve corresponds to air mass zero and is the spectrum of extraterrestrial solar radiation. The blackbody spectrum for 5762 K is also shown. The lowest two curves show the terrestrial spectrum for air mass 2, both with and without absorption. The difference between these two curves is emphasized by black in order to show how much absorption occurs at which wavelengths. The molecules responsible for these absorption bands are indicated. The spectral distribution is needed to calculate the performance of components whose behavior varies with wavelength. Examples are the efficiency of photovoltaic cells and the absorptance of selective coatings. Since the spectral distribution changes with atmospheric conditions and air mass, so do integrated solar properties such as photovoltaic cell efficiency and 'For very large air mass (i.e., large 02) the relation between m and 92 is more complicated. However the total energy incident at such large angles is so small that Eq. (3.1.1) is sufficiently accurate for practical applications. 2 This figure is based on an older value of the solar constant, but the effect on the spectral shape is negligible.
48
Insolation Data and Models
49
Figure 3.1.1 Spectral distribution of solar radiation for air mass 0 and 2 and 5762° K blackbody (From Kreith and Kreider [1978]).
absorptance of coatings. An exact system simulation would require as input solar spectra for all times of the day and of the year. The computational effort would be excessive, quite apart from the fact that such detailed data are rarely available. For most practical purposes one obtains an acceptable approximation by assuming a single constant spectrum. This has been confirmed with many numerical examples by Lind et al. [1980]. Therefore Lind et al. recommend a single standard spectrum for all solar energy calculations. It is reproduced here in Table 3.1.1. The spectral distribution of diffuse insolation does not seem to be as well known and documented as that for direct insolation. Since shorter wavelengths are scattered most, diffuse radiation spectra tend to be shifted towards shorter wavelengths [Kondratyev, 1969]. 3.2 TYPES OF INSOLATION The atmosphere causes not only absorption but scattering. Thus it is necessary to distinguish between beam insolation (coming directly from the solar disk), diffuse insolation, and hemispherical insolation (the sum of beam and diffuse, also called total or global). The magnitude of the solar flux received by a surface depends on its orientation. This necessitates a fairly elaborate notation to differentiate between various insolation types.
50
Active Solar Collectors and Their Applications TABLE 3.1.1 Standard Spectrum Recommended for Solar Energy Calculations byLindetal. [1980]a Percent
Wavelength (nm)
2.50 7.50 12.50 17.50 22.50 27.50 32.50 37.50 42.50 47.50 52.50 57.50 62.50 67.50 72.50 77.50 82.50 87.50 92.50 97.50
386.29 443.39 480.34 515.61 551.38 587.73 623.94 660.57 697.93 739.27 784.78 832.00 881.15 956.35 1023.73 1091.65 1208.54 1315.22 1624.27 2131.58
"The wavelengths correspond to the mean of the energy in each of the 20 equal energy intervals. Lind et al. also give the spectra with 50 and with 100 intervals.
We shall adopt the following conventions: 1. the subscript b designates beam insolation; the surface orientation is normal to the sun unless otherwise specified by a further subscript 2. the subscript d designates diffuse insolation; the surface orientation is horizontal unless otherwise specified by a further subscript 3. the subscript h designates hemispherical insolation; the surface orientation is horizontal unless otherwise specified by a further subscript To distinguish between power and energy we use the symbol / for irradiance [W/m2] while H is reserved for daily irradiation [J/m2] and q for monthly or annual irradiation [J/m2]. Since the hour is the shortest time interval for which insolation data are usually available, one commonly interprets hourly insolation data as average irradiance value during the hour. In the following chapters of this book almost all insolation values will be relative to the aperture plane of the collector. To minimize subscripts we therefore adopt the additional convention that /, H, and q without subscript designate the insolation incident on the aperture, with hemispherical insolation for the flat plate, beam insolation for collectors of high concentration, and beam plus 1/C diffuse for collectors of low concentration.
Insolation Data and Models
51
Let us illustrate this notation with some applications. First suppose that beam irradiance Ib is incident on the horizontal surface at zenith angle 6Z, while the diffuse irradiance is Id. Then the hemispherical irradiance Ih is
The daily total irradiation Hh on the horizontal surface is the integral of Ih from sunrise to sunset:
Next we turn to the irradiance 7 on a collector aperture. If the collector has a high concentration ratio (C> 10) then it accepts only a negligible amount of diffuse radiation. The effective irradiance on the aperture is therefore
if 6 is the angle of incidence of the sun on the aperture. For collectors of low concentration (e.g., the CPC) that do not receive radiation reflected from the ground, the expression is
because a collector of geometric concentration C accepts a fraction 1/C of the diffuse radiation, as shown in Subsection 5.3.3. Finally consider a flat plate tilted at an angle 0 from the horizontal. Since a flat plate accepts radiation from the entire hemisphere, it will also receive solar radiation reflected by the ground. Both the radiation from the ground and the diffuse sky radiation are usually treated as isotropic unless one has more precise information to the contrary. Being tilted, the collector accepts a fraction (1 + cos /3)/2 of the diffuse sky radiation and a fraction (1 — cos /3)/2 of the radiation Ihpffom& reflected by the ground, where pyom& is the reflectivity of the ground in front of the collector. Recommended values are pground = 0.2 without and 0.7 with snow, in the absence of better information. If the ground material in front of the collector is known, its reflectance can be looked up in Hunn and Calafell [1977]. If the incidence angle is 6 then the radiation seen by a flat plate collector is
Of course, only two of the three components Ib, Id, and Ih are independent, the third being fixed by Eq. (3.2.1). For example, if Id and Ih are given, one
52
Active Solar Collectors and Their Applications
can write Eq. (3.2.5) as
To calculate the performance of a solar energy system, one needs to know the diffuse and hemispherical irradiances for all times of the day and year. Such data are rarely available. What data are actually available and how one can supplement missing data by models form the subject of the remainder of this chapter.
3.3 INSTRUMENTS FOR MEASURING SOLAR RADIATION Most weather stations do not measure solar radiation but provide only some indirect estimate, for example, cloud cover or number of sunshine hours. The reporting of such ill-defined measures differs from station to station and the results are quite unreliable. Nonetheless they can be useful when calibrated against real measurements of solar radiation, and they have been widely used to supplement the existing network of measuring stations. As shown by Angstrom [Duffie and Beckman, 1980], the monthly average clearness index KT is closely correlated with the monthly average number of sunshine hours. An analogous linear correlation exists between KT and the monthly average cloudcover. However, the coefficients of these correlations vary from station to station, and the variation reflects not only climatic differences but differences in recording procedure. The instruments for measuring solar radiation are generally of two types: pyranometers, Fig. 3.3.1, which accept radiation from the entire hemisphere, and pyrheliometers, Fig. 3.3.2, which accept radiation from only one direction (more precisely from a cone of 2.8° half-angle). The pyranometer measures hemispherical insolation, while the pyrheliometer measures the beam component of the insolation. Both pyranometer and pyrheliometer operate on the principle of measuring the temperature rise of a black absorber as it
Figure 3.3.1
The Eppley
Precision Spectral
Pyranometer (Courtesy of Eppley Laboratories).
Insolation Data and Models
53
Figure 3.3.2 An Eppley Normal Incidence Pyrheliometer mounted on an altazimuth tracking mount (Courtesy Eppley Laboratories).
is heated by the sun. Their spectral response is uniform.3 High quality instruments (e.g., the Eppley Precision Spectral Pyranometer) can reach an accuracy of about ± 2% when carefully maintained and calibrated, and variations in sensitivity due to changes in incidence angle or tilt are small (on the order of 1 % or less). However, many of the instruments that have bee used have errors of 5% or more. In particular the pyranometers of the U.S. weather stations had been maintained rather poorly over the years and their calibrations drifted by unknown amounts, thus casting some doubt on the accuracy of the SOLMET data [SOLMET, 1978]. Most stations that measure insolation limit themselves to a horizontal pyranometer. Pyrheliometers, on the other hand, require careful maintenance, in particular, weekly tilt adjustments of the tracking axis; otherwise the data would be wrong and much worse than none at all. Some stations have attempted to avoid some of the maintenance problems of pyrheliometers by using pyranometers with shade rings (see Fig. 3.3.3); this, however, requires unreliable correction procedures and it does not eliminate the need for weekly adjustments. If one is interested only in instantaneous data (e.g., for collector testing), one can determine the beam radiation by holding an 3 Some low-cost insolation meters have been developed that use photovoltaic cells. Because of nonuniform spectral response their output can differ from pyranometers.
54
Active Solar Collectors and Their Applications
Figure 3.3.3 Pryanometer with shading ring to eliminate beam radiation (Courtesy Eppley Laboratories).
occulting disk of 0.1-m diameter 1 m above a pyranometer; this approximates the complement of the field of view of a pyrheliometer. In addition to pyranometers and pyrheliometers there are special purpose instruments; e.g., the circumsolar telescope. The circumsolar telescope was developed by the Lawrence Berkeley Laboratory [Hunt et al., 1977] in order to measure the solar radiation coming from the immediate vicinity of the solar disk. Such data are needed in order to analyze the performance of collectors with very high concentration because their acceptance angle can be considerably smaller than the field of view of the pyrheliometer. Therefore, pyrheliometer data overestimate the performance of collectors with very high concentration. The circumsolar telescope scans across a 6.4° arc with the sun at the center, and measures the brightness as a function of angle. Within 0.5° of the center of the sun the angular resolution is 1.5' of arc; outside this region it is 4.5' of arc. Building such an instrument is quite a challenge because of the extreme contrast between the brightness of the solar disk and the surrounding sky. On a clear day the intensity at the edge of the solar disk can drop 3 or 4 orders of magnitude in less than 0.1°. A circumsolar telescope is expensive to build and operate, and only four have been built. The effect of circumsolar radiation on collector performance is analyzed in Chapter 8 and circumsolar data are listed in Appendix E. 3.4 INSOLATION DATA
The number of stations that have records of measured insolation for more than 10 years of data is quite small. In the U.S. there are only 26 such stations with hourly radiation data, plus an additional 26 with daily insolation data. These stations are shown in Fig. 3.4.1. Their data are available on
Insolation Data and Models
55
Figure 3.4.1 The SOLMET solar radiation rehabilitation stations with hourly or daily pyranometer data (From Lunde [1980]. Reprinted with permission of John Wiley & Sons).
tapes and described in detail in the SOLMET User's Manual [SOLMET, 1978]. The data on these tapes have been obtained with a horizontal pyranometer. Horizontal pyranometer data are directly applicable only to horizontal flat plate collectors and solar ponds. For any other collector one must know something about the relative magnitudes of direct and diffuse radiation. This is the case even for flat plate collectors because the conversion of radiation data from the horizontal plane to the tilted collector plane is different for beam and for diffuse radiation. Unfortunately the data base for beam radiation has been very limited, consisting of a few years at a few stations. The SOLMET tapes do contain entries for beam radiation, but all of these beam "data" are calculated values based on a model by Randall and Whitson [1979] that was derived from about 12 years worth of data [Aerospace, 1976]. The SOLMET data are considered the best available and they have been widely used in recent years; e.g., most of the correlations in this book are based on these data. Nonetheless the user should beware of their limitations. The pyranometer data may have errors on the order of 5%, and the beam radiation "data" have significantly larger uncertainties. The deplorable state of the insolation data base is easy to understand. In the past there was hardly any interest in using direct solar radiation, and besides, the job of collecting reliable long term insolation data is tedious and unglamorous. Fortunately there is hope for improvement. In 1977 the U.S. network of insolation stations was expanded to include some 30 stations
56
Active Solar Collectors and Their Applications
with hourly pyrheliometer readings. These new data have not yet been analyzed (as of 1982). Data for hemispherical radiation on tilted surfaces are very rare. Such data are needed because on clear days diffuse radiation is not isotropic but slightly concentrated towards the sun. It seems that the popular assumption of isotropy for diffuse radiation leads to slight underpredictions of available radiation for typical flat plate applications by about 0-3% [Hay, 1979a, 1979b]. The most important insolation data are included in this chapter and in Appendix D. Figure 3.4.2 shows the annual total hemispherical solar radiation on the horizontal surface in a contour map for the U.S. Horizontal insolation data can, however, be quite misleading because most collectors are not horizontal. As will be shown in Chapter 11, the performance of most collector types is closely correlated with the average direct normal insolation. Therefore we add Fig. 3.4.3, a contour map for the annual daytime average beam irradiance ~Ib at normal incidence. Concentrating collectors miss part or all of the diffuse insolation; hence a look at the annual diffuse irradiation in Fig. 3.4.4 is instructive. It is worth noting that diffuse irradiation levels do not change much from one location to another. An extremely useful dimensionless presentation of insolation data is pro-
Figure 3.4.2 Yearly total hemispherical solar irradiation on a horizontal surface in GJ/m 2 (Adapted from SERI [1981]).
Figure 3.43 Yearly average beam normal irradiance [in kW/m] during daylight hours for the United States (Adapted form SERI [1982]).
Figure 3.4.4 Average annual diffuse solar irradiation on horizontal surface (in GJ/ m2) (Adapted from SERI [1981]).
57
58
Active Solar Collectors and Their Applications
vided by the clearness index KT, defined
where HQ is the extraterrestrial daily irradiation on a horizontal surface, given by Eq. (2.3.21) and listed in Table D.I. On heavily overcast days KT may be as low as 0.05-0.1 while on perfectly clear days it is in the range of 0.70-0.8. Monthly averages of KT range from 0.4 for very cloudy climates (e.g., Upstate New York) to 0.7 for very sunny climates (e.g., Albuquerque, NM). Many quantities of interest (e.g., beam insolation, diffuse insolation, and solar system performance) are closely correlated with KT. A contour map of annual average values A"7-iyear of the clearness index is shown in Fig. 3.4.5. Figure 3.4.6 gives an analogous map for the world. Monthly average values can be found in Appendix D. The reader may notice a serious inconsistency between Figs. 3.4.5 and 3.4.6. While the U.S. map [SERI, 1980] shows Kr values up to 0.70, the world map [WMO, 1980] suggests a maximum around 0.60, casting doubt on the quality of the data.
Figure 3.4.5 Mean annual clearness index K, for U.S. (KT = fraction of extraterrestrial radiation on a horizontal surface) (From SERI [1980]).
Insolation Data and Models
59
EXAMPLE 3.4.1
Estimate the daily hemispherical irradiation on a horizontal surface Hh on a clear and on a very cloudy day in December in New York City (latitude 41°N). SOLUTION
From Fig. 2.3.4 we find an extraterrestrial 24 h average irradiance of H0 = 13 MJ/m2. Taking a clearness index of KT = 0.70 for a clear December day in New York we find the irradiation Hh = 9.1 MJ/m2. With KT = 0.10 we find Hh = 1.3 MJ/m2 for a cloudy day.
3.5 INSOLATION MODELS 3.5.1
Types of models
The lack of sufficient insolation data necessitates the use of models. The type of model depends on what data are available and what data are needed. A large number of insolation models have been developed, and a brief guide might be helpful. Most insolation models fall into the following basic categories or are some combination of these categories. (i) Models based on cloud cover or sunshine hours These models calculate solar radiation when only cloud cover or hours of bright sunshine are known. The simplest examples are the linear correlations between KT and cloud cover or hours of sunshine [e.g., see Duffie and Beckman, 1980]. More sophisticated models combine cloud cover data with the calculated transmission of a clear atmosphere. Atwater and Ball [1978] have developed a detailed hourly model that takes as input atmospheric water content, cloud type and cover, and surface albedo; the accuracy is claimed to be ± 5%, better than most of the historical radiation measurements. If only monthly insolation values are needed then one can achieve comparable accuracy with a simpler model by Hay [1979a] that requires as input only bright sunshine hours and surface albedo. (ii) Models for atmospheric transmission Most of these models calculate the solar radiation for clear atmospheres only. Clear atmospheres are fairly easy to characterize, whereas clouds vary so much and in so many different ways that they can be treated at best in an approximate and average manner. Therefore these models are usually combined with some cloud cover data as mentioned in Section 3.5.1(i) above. However, calculations for clear sky conditions are also important in their own right. They are needed to assess the peak instantaneous and daily output of solar energy systems; such information is crucial for system opti-
Insolation Data and Models
61
mization. A particularly useful model for clear atmospheres will be presented in the following subsection. (Hi) Models to calculate beam and diffuse insolation from hemispherical insolation These models are needed to calculate the performance of all collectors that are not horizontal, when only the hemispherical horizontal insolation is known. Some of these models are described below, in Section 3.5.3, for models that need hourly or daily data and in Section 3.6 for long term averages. (iv) Models to calculate instantaneous or hourly insolation from daily totals These models are important because many stations have only data for daily total insolation. They are discussed in Section 3.6. (v) Models to calculate frequency distribution These models, needed for certain design tools, are discussed in Section 3.7. (vi) Other models for special applications As examples we mention models for circumsolar radiation and models for the angular distribution of diffuse sky radiation. A model that relates circumsolar radiation to atmospheric variables has been developed by Watt [1980]. As for the angular distribution of sky radiation, it is usually assumed to be isotropic. The assumption of isotropy makes the calculations simple and the accuracy is usually acceptable. When calculating long term average performance of solar energy systems in later sections of this book, we shall assume isotropy for the diffuse sky radiation. This assumption affects only flat plate collectors, and to a lesser extent collectors with very low concentration (e.g., the CPC). The error varies with weather and will result in underprediction of a few percent for the average radiation incident on tilted flat plate collectors. If greater accuracy is desired, the anisotropy model of Hay and Davies [1980] is recommended. Hay approximates the sky radiation as a linear combination of beam radiation and isotropic radiation. The relative weight of the beam and isotropic components is varied according to an anisotropy index, which is proportional to the atmospheric transmitFigure 3.4.6 Mean annual clearness index K, for the world. Annual global radiation at the earth's surface, expressed in percentage of the astronomically possible radiation (From World Meteorological Association [1980], with permission).
62
Active Solar Collectors and Their Applications
tance for beam radiation. Hay [1979b] has verified this model by comparison with a fairly extensive data base (hourly data on surfaces of tilts 0°, 30°, 60°, and 90°, both at Vancouver and at Toronto, with about 2 years of data each). When using insolation models one should keep their limitations in mind. Atmospheric phenomena are extremely complex and irregular. The simple correlations on which insolation models are based are not deterministic; rather they are true in an average sense only. For example, when the average daily clearness index K7 is 0.70, the ratio of diffuse over hemispherical radiation could be anywhere in the range from 0.10-0.50 and from the correlations we know only that the average is 0.25. Of course, if the data points exhibit such wide scatter, the average is not known with great accuracy. Therefore it is not surprising that many different correlations have been published and that some correlations for the same variables differ drastically from each other. While some of these discrepancies are due to errors in the data, some of them are due to inherent deficiencies of the model. Erbs [1980] presents a good discussion of this problem with regard to correlations between diffuse and hemispherical radiation. In some cases further investigations can reduce the uncertainties by including the effects of additional variables, but of course this makes the models more complicated. In the following we describe some of the most important correlations and models in detail. We believe they are the most useful and the most reliable that are now available. 5.5.2
Clear sky radiation
Hottel [1976] has presented a convenient method for estimating beam radiation under clear sky conditions. In terms of zenith angle 6Z and effective solar constant /0_eff, Eq. (2.2.1), the beam irradiance Ib at the earth's surface is indicated by4
The coefficients are
where A = elevation above sea level [km], and r0, r, and rk are the correction factors listed in Table 3.5.1 for different climates. "This model represents the atmosphere by a superposition of a gray gas (second term) and a black and a clear gas (first term).
63
Insolation Data and Models TABLE 3.5.1 Correction Factors for Climate Types for Clear Day Model of Hottel [1976]a Climate type
tropical midlatitude summer subarctic summer midlatitude winter
''o
23km visibility 0.95 0.97 0.99 1.03
5km visibility 0.92 0.96 0.98 1.04
r\
rk
0.98 0.99 0.99 1.01
1.02 1.02 1.01 1.00
This model is for an atmosphere with 23 km visibility and for altitudes A < 2.5 km. The altitude dependence of the coefficients is plotted in Fig. 3.5.1. For 5 km visibility the corresponding coefficients are given by
The clear sky diffuse irradiance Id (on the horizontal) can be estimated from a relation due to Liu and Jordan [I960]:
Figure 3.5.1 Constantsa?, af, and k* for the 23-km visibility standard atmosphere (As adapted from Hottel [1976] by Duffie and Beckman [1980]; reprinted with permission of John Wiley & Sons).
64
3.5.3
Active Solar Collectors and Their Applications
Correlations between beam, diffuse, and hemispherical radiation
In an average sense beam, diffuse, and hemispherical insolation are correlated with each other. To a good approximation the correlations for instantaneous insolation are independent of incidence angle if the solar radiation is expressed as fraction of extraterrestrial radiation. Two approaches have been used. One method correlates the ratio ljlh of hourly diffuse over hemispherical radiation with hourly clearness index
Both //, and Id are measured in the horizontal plane. An hourly time interval is short enough that the hourly irradiation is a good approximation for the instantaneous irradiance during the hour. Hence we use the symbols / for irradiance, even though the measured data are actually the irradiation during the hour. One such correlation [Erbs et al., 1982] is shown in Fig. 3.5.2 and parameterized by the curve fit
An alternative approach [Boes, 1980] correlates normal beam irradiance Ib with hemispherical horizontal irradiance //„ both expressed as fraction of extraterrestrial radiation. This correlation is shown in Fig. 3.5.3 and presented by the piecewise linear function
with coefficients A, and B, listed in the figure.
Figure 3.5.2 /,,//,, as a function of hourly clearness index (Adapted from Erbs etal. [1982]).
Insolation Data and Models
65
Figure 3.5.3 Percent of direct-normal radiation kb = hlL.cs as a function of percent of total horizontal radiation kr. The coefficients of the piecewise linear correlation kh — A,kr + BJ are also shown (From Boes[1980]).
A refined version of the correlation between Ib and lh has been developed by Randall and Whitson [1978]. It includes a statistical treatment of the fluctuations about the average, but it is so complicated that it requires a computer. It is currently considered the best model for this purpose and it has been used to add beam insolation "data" to all the SOLMET weather tapes. However, it has been derived from a rather small set of measurements (5 stations with a total of about 12 years of data, the best that was available at the time [Aerospace, 1976]). The above correlations require as input hourly data. If only daily totals are known, one can use a correlation analogous to Fig. 3.5.2, but with daily instead of hourly radiation values. This is shown in Fig. 3.5.4, and the corresponding curve fit is [Collares-Pereira and Rabl, 1979]
The solid line in this figure shows the curve fit to the data averaged over the year. The vertical lines indicate the rms deviations of individual data points about the yearly average. When the data are separated according to season (more precisely, according to sunset hour angle o>s), winter data tend to display lower diffuse ratios on clear days. This pattern is shown in Fig. 3.5.4 by the letters F, S, and W whose position shows the average Hd/Hh
66
Active Solar Collectors and Their Applications
Figure 3.5.4 Correlation of ratio of daily diffuse and daily hemispherical insolation with clearness index K,. Solid line shows curve fit of Eq. 3.5.12 for yearly average; the letters F, S, and W indicate seasonal variation (From CollaresPereira and Rabl [ 1979]). ratios for the seasons F = spring and fall (81.4° < ws < 98.6°), S = summer (98.6° > us), W = winter (w v < 81.4°). Correlations between beam, diffuse, and hemispherical radiation exhibit a great deal of scatter. Some reduction of this scatter has been achieved by Hay [1976], who includes the effects of multiple reflections between the earth's surface and the atmosphere. However, other causes of scatter remain [Erbs 1980]. 3.6 LONG TERM AVERAGE INSOLATION For many applications one needs to know the long term average monthly irradiation on tilted or tracking surfaces. Before describing a simple algorithm for this purpose, we want to make a general comment on the role of long term averages and monthly averages in solar energy. Obviously one cannot predict the performance of a solar energy system for a particular day of a particular year (e.g., the 14th of July 1989) nor does one need to. After all, the economics is determined by the energy that the system delivers during its lifetime, not by the performance on a particular day. While the yearto-year fluctuations of the weather are large and unpredictable, the long term average is known from historical data and serves as basis for solar energy calculations. In some applications the seasonal variations of insolation and load neces-
Insolation Data and Models
67
sitate separate calculations for different parts of the year. In such a case one could calculate the long term average insolation for every day of the year. For example, to calculate the long term average insolation for the 14th of July, one would use all available weather data for this date during past years. In practice, one can simplify this task by a factor of 30 if one performs one calculation for the central day of each month. This is permissible because the change of the long term average weather during the course of one month is sufficiently small. In principle, a month is a somewhat arbitrary interval but it is extremely convenient in practice because monthly data are readily available from many sources. The problem of year-to-year variations also arises if one uses real hourly weather data. For the SOLMET tapes the following approach has been taken. First the long term monthly average values of insolation, temperature, humidity, windspeed etc., were calculated. Then a so-called typical meteorological year (equals TMY) was spliced together by selecting those months that agree most closely with the long term averages. The algorithm described in this section determines the long term average irradiance on any surface at any time of day or year. Only one input parameter is required, the average clearness index KT for the month (or equivalently, the monthly average hemispherical insolation Hh on the horizontal surface, since H0 during a month can be approximated by a constant). __As first step one calculates the monthly average daily diffuse irradiation 77 rf from Fig. 3.6.1 or Eq. (3.6,1) [Collares-Pereira and Rabl, 1979]:
This equation is the long term average analogue of Eq. (3.5.12); it differs from Eq. (3.5.12) because it depends on the frequency distribution of clear
Figure 3.6.1 Correlation for long term average values, Hd/Hh, versus K, and
68
Active Solar Collectors and Their Applications
and cloudy days. A point on the Hd/Hh curve is obtained by averaging Hd/ Hh data corresponding to a fixed value of Kr, whereas a point on the TJd/TJh curve is the result of averaging both Hd/Hh and KT over all days of the month. As second step, one converts the daily irradiation totals Hd and Hh into the instantaneous irradiance values 1d and Th by means of the correlations
where
In these equations rday = length of day = 24 h, and the hour angle
represents the time of day t, while the time of year enters only through the sunset hour angle u>s
The coefficients a and b are given by
The correlation functions rd and rh are plotted in Fig. 3.6.2. Like the other correlations in this section, they are reliable only in an average sense5; the ratios Id/Hd and Ih/Hh for any particular day can be quite different. The units of rd and rh in Fig. 3.6.2 are in h ' on the left side, corresponding to the choice' of h for rday = 24 h in Eqs. (3.6.4) and (3.6.5). This gives ~Id and ~Ih the units of J/h X m2. With rday = 24 X 3600 sec, rd and rh have units of sec"1 as shown on the right side; this yields 7d and lh in W/m2. Of the three insolation types (beam, diffuse, and hemispherical) only two are independent. Having determined the average diffuse and hemispherical irradiance for a particular time of the day and the year, one knows also the 5 The perceptive reader may notice that this fit for rh is slightly inconsistent with the normalization condition that the integral of rk from sunrise to sunset be unity. This discrepancy is small (about 1%) and not significant in practice.
Insolation Data and Models
69
Figure 3.6.2 Correlation between daily total solar irradiation and instantaneous solar irradiance. (a) For hemispherical insolation rh = lh/Hh. (b) For diffuse insolation ra — ld/Hd. The units of rd and rh are h~' on left and 10~6 sec"1 on right side (From Collares-Pereira and Rablf 1979]).
beam irradiance. The beam irradiance at normal incidence is
where 6, is the solar zenith angle. Thus, the average direct and diffuse insolation on any surface and for any time of day can be calculated if one knows the monthly average clearness index KT. The model can be used to calculate the solar contribution to heat-
70
Active Solar Collectors and Their Applications
ing and cooling loads of buildings as well as the energy incident on solar collectors of various kinds. Explicit expressions for the integrated daily total irradiation incident on the most important fixed or tracking surfaces are listed in Chapter 11. It is instructive to show why the algorithm of this section should predict the long term average radiation correctly by its very construction. As stated above, with the isotropy assumption for the diffuse component, the long term average insolation on any aperture is uniquely determined by JH and ld. These in turn can be written in a form that shows explicitly the long term average as a summation over a period N days:
After multiplying and dividing by
this can be rearranged as
using the definition of rh, Eq. (3.6.3). The analogous statement for Id is
Therefore the long term average insolation must be correct if rh, rd, and Hd/ Hh are correct. The validity of the algorithm is guaranteed to the extent to which these correlations can be described by location independent curve fits. In practice the accuracy appears to be on the order of 3% as far as the average monthly irradiation on solar collectors is concerned [Collares-PereiraandRabl, 1979]. EXAMPLE 3.6.1
Calculate the long term average insolation on an east-facing vertical wall in Rome at equinox at 8:00 in the morning, if the long term average hemispherical insolation is Hh = 11.96 J/m 2 . The latitude is X = 41.8°.
Insolation Data and Models
11
SOLUTION
The sunset hour angle is u>s — 90° and the hour angle is o> = —60°. The irradiance on the wall is, from Eq. (3.2.6) with j3 = 90°,
with incidence angles given by Eq. (2.3.5) for cos 9Z = cos X cos d cos o> + sin X sin <5 and by cos 0wai, = —cos 5 sin w from Table 2.4,1, flat plate entry with /? = 90° and <£ = — 90°.^Intermediate steps of the calculation yield HQ = 27.94 MJ/m2, KT = 0.43, TJd = 5.72 MJ/m2, rd = 0.064 h~\ rh = 0.056 tr1, Id — 102 W/m2, and //, = 186 W/m2. Assuming the ground reflectance to be Pground = 0.2 one obtains the result
for the average hemispherical irradiance on the wall at this time of day and year.
3.7 FREQUENCY DISTRIBUTION OF DAILY RADIATION VALUES
For collectors with heat loss the monthly average insolation on the aperture does not determine the energy delivery completely. To explain this fact, consider a collector whose heat loss equals the monthly average insolation absorbed by the collector. If the insolation were always constant, this collector would not deliver any useful energy. If, on the other hand, the month consisted of completely clear days and of completely cloudy days (with a distribution that results in the same monthly average as before), then this collector would deliver useful energy on the clear days. Therefore the performance of solar thermal collectors depends not only on the average insolation but also on the fluctuations about the average. This effect can be analyzed if the frequency distribution of insolation values is known. As was first shown by Liu and Jordan [1963], the frequency distribution can be approximated by a universal distribution function, the cumulative frequency distribution for the clearness index
The cumulative frequency distribution f(K-,), also called fractional time distribution, is defined such that
72
Active Solar Collectors and Their Applications
If one knows the functional form of J(KT), one can calculate how many hours of the month or year the insolation is above or below a specified value. Take, for example, a location at 40° latitude in December. The corresponding daily extraterrestrial irradiation on a horizontal surface is
from Fig. 2.3.4. If we want to know how many days of the month the actual irradiation Hh is less than 3.0 MJ/m2, we simply evaluate the function f(KT) at the value
As shown by Liu and Jordan [1963], the frequency distribution of hourly radiation values, when expressed in terms of hourly clearness index
is nearly independent of time of day and can be approximated by the distribution for the daily clearness index Kr = Hh/H0. Furthermore the cumulative frequency distribution is nearly independent of location and of time of year; the only significant dependence lies in the long term average value KT of the clearness index. Recently this universal fractional time distribution curve has been rederived from data for 90 stations in the U.S. with approximately 20 years for each station [Bendt et al., 1981]. The resulting distribution f(KT, ~KT) is shown by the solid lines in Fig. 3.7.1 for KT = 0.3, 0.4, 0.5, 0.6, and 0.7. A remarkably good fit to these data is provided by the function, shown by dotted lines in Fig. 3.7.1,
which corresponds to random insolation sequences. In this equation ATrmin and -^r>max are the smallest and largest values of the clearness index, with fractional times 0 and 1, respectively,
Kr,mm corresponds to heavily overcast days and takes approximately the same value
for all locations. ATr,max represents the clearest days and depends somewhat on climate through the average clearness index KT. From Fig. 3.7.1 one sees
Insolation Data and Models
73
Figure 3.7.1 Comparison between fractional time distribution of insolation data and fractional time distribution for random insolation sequences (From Bendtetal. [1981]). that KT^mm ranges from 0.69 for very cloudy climates (KT = 0.3) to 0.81 for very clear climates (Kj- - 0.7). The constant 7 in Eq. (3.7.4) is determined uniquely in terms of KT, KT,mm, and -Kr>max by the equation
It is a transcendental equation that can be solved by iteration. Returning to the example at the beginning of this section, let us suppose that the climate has an average clearness index ~KT = 0.5. Then the fraction of time that Hhis below 3.0 MJ/m2 is found from Fig. 3.7.1 to bef(KT = 0.23, KT = 0.5) = 0.15. In other words, during 0.15 X 31 days = 4.65 days of December, the daily hemispherical irradiation on the horizontal is Hh < 3.0 MJ/m2. 3.8 YEARLY IRRADIATION ON APERTURE For horizontal collectors such as solar ponds the yearly total radiation is given directly in standard insolation tables and figures, for example, in Fig. 3.4.2 and in Appendix D. For all other collector types the radiation on the aperture depends on collector tilt or tracking mode, on concentration ratio, on shading effects and incidence angle modifiers, and on the relation between beam and diffuse insolation. However, as shown by Rabl [1981], the yearly radiation on the aperture is closely correlated with yearly average beam insolation. For some collectors, in particular the flat plate, there is also a dependence on geographic latitude. The correlation is shown in Fig. 3.8.1 (For a discussion of the underlying assumptions, see Section 11.3.) To use it, one first needs the yearly average
74
Active Solar Collectors and Their Applications
Figure 3.8.1 Yearly irradiation q0 [GJ/m2] incident on aperture of principal collector types. These correlations include typical incidence angle modifiers, as explained in Section 11.3. (Curve for central receiver is based on one specific design at 35° latitude only, and includes intercept factor.) This graph shows only radiation availability, not system output.
beam irradiance at normal incidence 7A during daylight hours. For the U.S. this can be read directly from the contour map in Fig. 3.4.3. For other areas 7ft can be determined from the correlation
which is displayed in Fig. 3.8.2, between I,, and the annual average clearness index K,:
The average beam normal irradiance /,, during daytime is the abscissa (x axis) in Fig. 3.8.1; the corresponding annual irradiation q0 incident on the aperture of parabolic dish, parabolic trough, and flat plate can then be read directly from the ordinate (y axis) in GJ/m 2 per year. This figure illustrates the relative importance of 2-axis tracking and of diffuse radiation. The loss of diffuse radiation puts the parabolic dish at a disadvantage in cloudy locations. In clear climates, on the other hand, the diffuse radiation is less
Insolation Data and Models
75
Figure 3.8.2 Correlation between yearly average clearness index ~K, and yearly
average daytime beam normal irradiance 7h (From Rabl [1981]).
important than the incidence angle, and the parabolic dish surpasses the flat plate in collectible energy. For collectors not included in Fig. 3.8.1, one finds q0 as the value of q(x) at zero threshold X = 0 from the correlations in Section 11.3.2. The correlation of the annually collectible solar radiation on concentrating and tilt equal latitude flat plate collectors with the intensity of beam insolation leads to some surprising results. For example, most people think of Florida as a place with plentiful sunshine, yet its level of beam normal insolation is about as low as in Minnesota and barely more than half of what is available in New Mexico. Florida is much warmer than Minnesota because the hemispherical insolation on the horizontal surface is much higher closer to the equator but its southerly location does not give the "sunshine state" an advantage in annual average beam normal insolation. Generally desert regions have the highest levels of beam insolation. In the tropics beam insolation is reduced by clouds and high humidity. 3.9 SEASONAL VARIATION OF INSOLATION In addition to the total annual solar radiation one may also want to know something about its seasonal variation. This is particularly important if the load varies with time of year, for example, for heating and cooling. The seasonal variation of insolation is due to the following factors: (i) the seasonal variation in the elevation of the sun above the horizon and in the length of the day (ii) seasonal changes in the average transparency of the atmosphere (iii) the ± 3% variation of the effective solar constant To give the reader some intuition for the seasonal variation we present in
Figure 3.9.1 Variation of monthly average insolation on a horizontal surface (normalized to an annual average value of unity) (Adapted from Rabl and von Hippel[1983]).
Figure 3.9.2 Variation of monthly average insolation on a south-facing vertical surface (normalized to an annual average of unity) (Adapted from Rabl and von Hippel[1983]).
Figure 3.9.3 Variation of monthly average insolation on a south-facing collector tilted at latitude — 5" (normalized to an annual average of unity) (Adapted from Rabl and von Hippel [1983]). 76
Insolation Data and Models
77
Figs. 3.9.1-3.9.6 some results for the principal collector orientations. Derived as an average over all 26 SOLMET stations of the U.S., these results are typical of midlatitudes (25-50° north) and of a wide range of climates. For each collector type we have calculated the monthly irradiation incident on the aperture q(s, m) for month m and station s. Then we have normalized the monthly values by dividing them by the yearly total irradiation q(s) for that station. These normalized insolation values have the same yearly average of unity for all stations and are therefore useful for exhibiting the general pattern of seasonal variations about the average. Of course, different stations have somewhat different seasonal behavior, but showing the numbers for each of the 26 stations would be more confusing than instructive. Therefore we decided to show only the mean for all stations and, as a measure of the spread around this mean, the standard deviation. The crosses in Figs. 3.9.1-3.9.6 indicate the means /x(m)
for each month m. The standard deviation a(m) of the individual stations about the station mean is calculated according to
and is indicated by the length of the vertical bars above and below these crosses. Figures 3.9.1-3.9.6 cover six collector orientations, three fixed and three tracking: (i) fixed horizontal flat plate (ii) fixed south-facing vertical flat plate
Figure 3.9.4 Variation of monthly average direct beam insolation on a 2-axis tracking collector (normalized to an annual average of unity) (Adapted from Rabl and von Hippel [1983]).
78
Active Solar Collectors and Their Applications
Figure 3.9.5 Variation of monthly average insolation on a parabolic trough collector with horizontal east-west tracking axis (normalized to an annual average of unity) (Adapted from Rabl and von Hippel [1983]).
(iii) fixed south-facing flat plate with tilt equal latitude (iv) concentrator tracking about horizontal east-west axis (v) concentrator tracking about horizontal north-south axis (vi) concentrator tracking about two axes The horizontal flat plate and the concentrators with north-south tracking axis exhibit the largest seasonal variation, ranging from about 0.5 of the annual average in winter to 1.5 of the average in summer. The output is most uniform for the vertical flat plate (0.75-1.15) and for the concentrator with horizontal east-west tracking axis (0.7-1.3). Of course these figures say nothing about the absolute radiation levels for the various collector types, nor do they take into account the effect of thresholds due to collector heat loss. For example, in midlatitudes the concentrator with north-south tracking axis receives on the order of 10% more yearly radiation than the one
Figure 3.9.6 Variation of monthly average insolation on a parabolic trough collector with horizontal north-south tracking axis (normalized to an annual average of unity) (Adapted from Rabl and von Hippel [1983]).
Insolation Data and Models
79
with east-west axis, as can be seen from Fig. 11.3.8. For a detailed discussion of the seasonal variation the reader is referred to Rabl and von Hippel [1983]. 3.10 SUMMARY This chapter is fairly long, and the plethora of different insolation types may be confusing. We therefore present a brief summary of the nomenclature and of the insolation models. The symbol / designates irradiance (instantaneous or hourly average power, in W/m2 or J/hr X m2) while H designates irradiation (daily energy, in J/m2), with subscripts for insolation type: 70 = solar constant = 1373 W/m2
Ib
beam irradiance at normal incidence
I d
diffuse irradiance on horizontal
Ih
hemispherical irradiance on horizontal
H0
daily extraterrestrial irradiation on horizontal
Hd - daily diffuse irradiation on horizontal Hh = daily hemispherical irradiation on horizontal / and H without subscripts designate the irradiance and irradiation on the aperture of the collector. Bars indicate long term averages. The insolation models are summarized in Table 3.10.1. For the calculations several dimensionless quantities are important. The clearness index is the ratio of terrestrial and extraterrestrial daily irradiation:
The time of day t is expressed as hour angle u
80
Active Solar Collectors and Their Applications
TABLE 3.10.1 Summary of Insolation Models Input
Output
Correlation
Clear days only Atmospheric visibility; elevation above sea level; cos 0Z; 70ieff
7b
Eqs. (3.5.1) to (3.5.7), Fig. 3.5.1 and Table 3.5.1
4; 4,eff
4
Eq. (3.5.8)
Diffuse insolation for any particular hour or day 7h; cos 6,; 70eff
4
7h; cos 0Z; 70.efT
4 /4
H\,\ HO
Eqs. (3.5.9) and (3.5.10) or Fig. 3.5.2 Eq. (3.5. 11) and Fig. 3.5. Eq. (3.5. 12) or Fig. 3.5.4
Long term average insolation
74; HO; a.
/4
Eq. (3.6.1) or Fig. 3.6.1
HA, w; ws
4 4
Eq. (3.6.2) or Fig. 3.6.2a
HI,', w; «>s
Eq. (3.6.3) or Fig. 3.6.2b
Frequency distribution Hh', Hf>
Distribution of 77,, values
Eq. (3.7.4) or Fig. 3.7.1
Yearly irradiation ^T.yr
4
4
Eq. (3.8.1) or Fig. 3.8.2
g0 = irradiation on aperture
Fig. 3.8.1 (also Figs. 11.3.1-11.3.7)
with rday = 24 h, and the time of year enters through the sunset hour angle o)5 given by
where X is the latitude and d is the declination, given by Eq. (2.2.4).
REFERENCES Aerospace. 1976. Report No. ATR-76 (7523-ll)-9. El Segundo, CA: The Aerospace Corporation. Atwater, M. A. and Ball, 3. T. 1978. "A Numerical Radiation Model Based on Standard Meterological Observations." Solar Energy 21:163. Bendt, P., Collares-Pereira, M., and Rabl, A. 1981. "The Frequency Distribution of Daily Insolation Values." Solar Energy 27:1.
Insolation Data and Models
81
Boes, E. C. 1980. "Fundamentals of Solar Radiation." In Solar Energy Handbook, Kreider, J. F. and Kreith, F., editors. New York: McGraw-Hill. Collares-Pereira, M. and Rabl, A. 1979. "The Average Distribution of Solar Radiation—Correlations Between Diffuse and Hemispherical and Between Hourly and Daily Insolation Values." Solar Energy 22:155 (1979). Duffie, J. A. and Beckman, W. A. 1980. Solar Engineering of Thermal Processes. New York: John Wiley & Sons. Erbs, D. G. 1980.' "Methods for Estimating the Diffuse Fraction of Hourly, Daily, and Monthly—Average Global Solar Radiation." M.Sc. thesis. Madison, WI: Dept. of Mech. Engineering, University of Wisconsin. Erbs, D. G., Klein, S. A, and Duffie, J. A. 1982. "Estimation of the Diffuse Radiation Fraction for Hourly, Daily and Monthly—Average Global Radiation." Solar Energy 28:293. Hay, J. E. 1976. "A Revised Method for Determining the Direct and Diffuse Components of the Total Shortwave Radiation." Atmosphere 14:278. Hay, J. E. 1979a. "Calculation of Monthly Mean Solar Radiation for Horizontal and Inclined Surfaces." Solar Energy 23:301. Hay, J. E. 1979b. "Study of Shortwave Radiation on Non-Horizontal Surfaces." Canadian Climate Center Report 79-12. Downsview, Ontario: Atmospheric Environment Service. Hay, J. E. and Davies, J. A. 1980. "Calculation of the Solar Radiation Incident on an Inclined Surface." In Proceedings First Canadian Solar Radiation Data Workshop, Toronto 1978, J. E. Hay and T. K. Won, editors. Published by Minister of Supply and Services, Canada. Hottel, H. C. 1976. "A Simple Model for Estimating the Transmittance of Direct Solar Radiation Through Clear Atmospheres." Solar Energy 18:129. Hunn, B. D. and Calafell, D. O. 1977. "Determination of Average Ground Reflectivity for Solar Collectors." Solar Energy 19:87. Hunt, A. J., Grether, D. F., and Wahlig, M. 1977. "Techniques for Measuring Circumsolar Radiation." Report LBL-8345. Lawrence Berkeley Laboratory. Kondratyev, K. Y. 1969. Radiation in the Atmosphere. New York: Academic Press. Kreith, F. and Kreider, J. F. 1978. Principles of Solar Engineering. New York: McGraw-Hill. Lind, M. A., Pettit, R. B., and Masterson, K. D. 1980. "The Sensitivity of Solar Transmittance, Reflectance, and Absorptance to Selected Averaging Procedures and Solar Irradiance Distributions." ASME /. Solar Energy Eng. 102:34. Liu, B. Y. H. and Jordan, R. C. 1960. "The Interrelationship and Characteristic Distribution of Direct, Diffuse, and Total Solar Radiation." Solar Energy 4:1. Liu, B. Y. H. and Jordan, R. C. 1963. "A Rational Procedure for Predicting the Long Term Average Performance of Flat Plate Solar Energy Collectors." Solar Energy 7:53. Lof, G. O. G., Duffie, J. A., and Smith, C. O. 1966. "World Distribution of Solar Radiation." Report No. 21. Madison, WI: Solar Energy Lab., Univ. of Wisconsin. NASA. 1971. "Solar Electromagnetic Radiation." NASA Report SP-8005. National Aeronautics and Space Administration. Orgill, J. F. and Hollands, K. G. T. 1977. "Correlation Equation for Hourly Diffuse Radiation on a Horizontal Surface." Solar Energy 19:357. Rabl, A. 1981. "Yearly Average Performance of the Principal Solar Collector Types." Solar Energy 27:215. Rabl, A. and von Hippel, F. 1983. "The Solar Radiation Resource." Energy 8:295.
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Active Solar Collectors and Their Applications
Randall, C. M. and Whitson, M. E. 1978. "Final Report, Hourly Insolation and Meteorological Data Bases Including Improved Direct Insolation Estimates." Report ATR-78 (7592-1). El Segundo, CA: Aerospace Corporation. SERI. 1980. "Insolation Data Manual." Report SERI/SP-755-789. Solar Energy Research Institute. SERI, 1981. "Solar Radiation Energy Resource Atlas of the United States." Report SERI/SP-642-1037 (1981). Golden, CO: Solar Energy Research Institute. SERI. 1982. "Direct Normal Solar Radiation Data Manual." Report SERI/SP-2811658. Solar Energy Research Institute. SOLMET. 1978. Volume 1: User's Manual, Volume 2: Final Report, Hourly Solar Radiation Surface Meteorological Observations. Report #TD-9724. Asheville, NC: National Climatic Center. Thekaekara, M. P. 1974. Supplement to the Proceedings of the 20th Annual Meeting of the Institute for Environmental Science, p. 21. "Data on Incident Solar Energy." Watt, A. D. 1980. "Circumsolar Radiation. Report SAND80-7009. Sandia National Laboratories. WMO. 1980. "Report on World Insolation Data." World Meteorological Organization.
4. COLLECTOR EFFICIENCY AND COLLECTOR TESTING
The sun also shines on the wicked. Seneca
4.1 DEFINITION OF INSTANTANEOUS COLLECTOR EFFICIENCY
It is natural to define the instantaneous efficiency 77 of a solar thermal col lector as the ratio of the useful heat Q [in W] delivered per aperture area A [in m2] and the insolation / [in W/m2], which is incident on the aperture1
The useful heat Q is related to flow rate m [in kg/sec], specific heat at constant pressure c [kJ/kg °K], and inlet and outlet temperatures Tin and Tom by
The efficiency may depend on many factors; e.g., collector temperature, ambient temperature, insolation, flow rate, and incidence angle. In order to characterize a collector, one must therefore specify carefully the conditions under which the efficiency has been measured or calculated. One would like to specify the conditions in such a way that the efficiency is defined unambiguously and can be measured reproducibly. This can be accomplished most easily if one bases the efficiency on clear sky conditions. Even though clear skies are not identical, they are sufficiently reproducible to provide a useful reference for collector testing.2 'One can base the efficiency on net area or on gross area of the collector, provided one follows the same convention consistently, including the reporting of collector cost. Throughout this book A is understood to be the net or aperture area, and all efficiencies and energy flows are based on aperture area. Note that the ASHRAE [1977] test procedure is based on gross area. 2 As an alternative some people have advocated the use of an all-day operating efficiency. However, the all-day efficiency depends on the weather and on the incidence angles during the test day. It is not clear how to interpret such data and predict from them the performance for different days and locations.
83
84
Active Solar Collectors and Their Applications
4.1.1 Specification of insolation For concentrating collectors a troublesome ambiguity with Eq. (4.1.1) arises from the mismatch between the angular acceptance characteristics of the collector on the one hand and of the insolation measuring instrument on the other. One might like to base the efficiency on the irradiance 7in accepted by the collector. However, this turns out to be impractical. With respect to their angular acceptance, all insolation meters that are or have been generally available fall into one of two classes: 1. the pyranometer for measuring the hemispherical irradiance //,. Its field of view is hemispherical; i.e., its acceptance half-angle 6a = 90°. 2. the pyrheliometer for measuring the beam irradiance Ib. Its field of view is approximately a cone of half-angle 6a = 2.5°, 10 times the angular radius of the sun. These properties are sketched in Figs. 4.1.1 a, 4.l.lb, 4.l.lc, and 4.1.Id. The CPC in Fig. 4.l.lc has an acceptance half-angle much larger than 2.5°, and the proper value of 7in lies between Ib and Ik. The focusing parabola in Fig. 4.1.Id, on the other hand, has a much smaller acceptance angle than the pyrheliometer and 7in is smaller than Ib. The difference between Ib as measured by a pyrheliometer and the radiation from the solar disk itself is called circumsolar radiation (See Section 8.6). The variability of clouds and haze can cause significant scatter in the
Figure 4.1.1 Typical acceptance angles: (a) pyranometer, 26a = 180°; (b) pyrheliometer, 26a = 5°; (c) nontracking CPC, 26„ = 60°; (d) tracking parabola, 26 a = 1°.
Collector Efficiency and Collector Testing
85
ratios LJIh and I-JIh for a particular collector; this will be reflected in the efficiency data if they are based on Ib or //,. One could reduce or eliminate this scatter by designing solar radiometers with the same angular acceptance properties as the collector to be tested, but that would be costly. Also, it is desirable, in the interest of facilitating comparison between different collector types, to report all efficiencies with respect to either lh or Ib. In principle, the choice between Ih and Ib is somewhat arbitrary, but in order to minimize scatter of data points due to variable atmospheric conditions, the following convention is recommended: The efficiency of flat plate collectors should be based on Ih (pyranometer), whereas that of tracking concentrators should be based on Ib (pyrheliometer), as has been done traditionally. For concentrators with low concentration, for example CPCs, the situation is less clear and no consensus has emerged as yet. If the diffuse radiation is isotropic, then a fraction 1/C is accepted by a collector of geometric concentration C as shown in Section 5.3.3. Hence it is reasonable to base the efficiency Eq. (4.1.1) on the insolation
We have assumed this rule for the long term performance predictions in Chapter 11. For collectors with concentration C > 10 the contribution of diffuse radiation can be neglected. As an alternative to Eq. (4.1.3) one could base the insolation on Ih if C < 2 and on Ib if C > 2. To avoid misunderstandings, it is advisable to indicate clearly by subscripts which type of insolation has been assumed. This makes it possible to convert to a different insolation base. For example, if the efficiency of a focusing collector has been reported with respect to a pyranometer as
one can convert it to equivalent pyrheliometer data
with the conversion factor
This effect is not negligible since even on a clear day Ib will be at most 90% of I,. As another example, suppose the efficiency of a CPC with concentration
86
Active Solar Collectors and Their Applications
C — 1.5 has been based on Ih. Then the efficiency would be
if one wants to base it on the insolation 7i0iC within the acceptance angle, Eq. (4.1.3). The conversion from r\h to rj i n c is
Rabl et al. [1980] show how these formulas can be used to minimize the scatter of CPC collector test data due to varying atmospheric conditions. In the following we shall drop the subscripts for efficiency and insolation, whenever the equations hold equally for all collector types, and we shall assume the above conventions for insolation base. EXAMPLE 4.1.1
Figure 4.4.4 shows the efficiency % relative to beam irradiance 7b for several collectors, including a CPC of concentration C = 1.5. How would the efficiency change if it were based on hemispherical irradiance 7h or on irradiance 7in c within acceptance angle, assuming that 7d = 150 W/m2 and 7b = 850 W/m2 during the test. SOLUTION Relative to 7h the efficiency is
Relative to 7 inC the efficiency is, from Eq. (4.1.8),
4.1.2
Optical efficiency and U value
The optical efficiency 770 is defined as that fraction of the solar radiation I which reaches the receiver and is absorbed there:
For many collectors it is easy to derive approximate formulas for the optical efficiency. For example, in a flat plate collector with a single cover of trans-
Collector Efficiency and Collector Testing
87
missivity T and with an absorber of absorptivity a the optical efficiency is
relative to hemispherical insolation, if all rays reach the absorber and if multiple reflections between cover and absorber are neglected. For a parabolic trough with reflectivity p, transmissivity r of the receiver glazing, and absorptivity a of the absorber, the optical efficiency is approximately
relative to beam insolation. For more accurate formulas, see Eqs. 5.2.20 and 8.4.7. Multiple reflections between absorber and cover can raise the optical efficiency by about 1%. On the other hand, the intercept factor, defined as the fraction of the incident rays that reach the receiver, may be as much as 5%-10% less than unity. These corrections are very design dependent. In the flat plate collector literature 7j0 has also been called the TO. product. In most collectors TJO is the collector efficiency ri(0) at AT = 0, AT = Tabs — Tamb being the temperature difference between absorber and ambient. In some collectors, however, especially in nonevacuated collectors with reflectors adjacent to the absorber, the solar radiation absorbed in the reflector may raise the reflector temperature above the absorber temperature if the latter is near ambient. In that case the receiver can gain thermal energy from the reflector and ?j(0) can be larger than TJO. In certain collectors 7?(0) can exceed the optical efficiency 7j0 by several percent. ij(0) is called the effective optical efficiency, and formulas for it can be found in Section 9.10. The difference between 7j(0) and TJO is of interest only when one attempts to calculate the efficiency from first principles. In this chapter we are concerned with measurements of the efficiency curve, and we need not worry about the origin of ?j(0). Hence, we shall not distinguish between the effective optical efficiency 77(0) and the true optical efficiency T?O in this chapter. When the receiver temperature is above ambient and the collector loses heat at a rate
In order to exhibit the temperature dependence of the heat loss it is convenient to define the collector heat loss coefficient or U value as
where rabs is the absorber surface temperature and Tamb is the ambient temperature. The U value depends only weakly on temperature, and in most cases one gets an excellent approximation if one chooses a constant U value that cor-
88
Active Solar Collectors and Their Applications
responds to typical operating conditions. The resulting linearized efficiency equation
is convenient because it characterizes the performance of a solar collector in terms of only two parameters, rj0 and U. If the efficiency is plotted versus the variable (Tabs — Tamb)/I, one obtains a straight line with intercept ??0 and slope — U, as sketched in Fig. 4.1.2a. Strictly speaking, the U value is not constant but depends on receiver temperature, air temperature, sky temperature, wind velocity, and collector tilt. In practice most of these variables have only a small effect, and, besides, their values under actual operating conditions generally will not be known with sufficient accuracy to justify the effort of calculating their effects accurately. The ASHRAE test procedure [ASHRAE, 1977], for example, does not account for the effects of wind speed and of differences between air and sky temperature.3 When radiative losses are relatively small, the temperature dependence of the U value tends to be mild. This is the case for flat plate collectors with 3
For a discussion of the effect of wind speed on U values, see, for example, Tabor [1978].
Figure 4.1.2 Instantaneous collector efficiency curve, (a) U value independent of temperature, (b) U value increasing with temperature. Solid line shows exact curve; dashed line shows approximation by linear equation with intercept J70jCquiv and slope — C/equiv which is equivalent over the range of temperature and insolation values in a specific application.
Collector Efficiency and Collector Testing
89
selective coatings. In evacuated collectors, on the other hand, the temperature dependence of U can be quite significant. Above all, the variation of U depends on the range of temperatures of interest. Some flat plates are tested and operated at such low temperatures that the test results are not even accurate enough to detect changes in U value. High temperature collectors, on the other hand, may be operated over temperature ranges of several hundred degrees, and it may be necessary to keep track of the temperature dependence of U. To the extent that U is not constant, the efficiency depends on Tabs, Tamb, and on / separately, and it is no longer possible to plot the efficiency as a single line versus the single variable (rabs — ramb)//. If one wants to include a second order correction, one can expand the U value as
Tabor [1978] has analyzed flat plate data with a different formula. Writing
where U0 and p are constants chosen to fit the data, he found that the exponent p fell in the range of 0.1-0.3. Unfortunately, some people have used an expansion of the form
blithely ignoring the fact that a heat transfer coefficient does not depend on solar radiation. Furthermore this conceptual error causes problems with the interpretation of the heat transfer factor Fin in Eq. (4.2.3), since Eq. (4.2.4) for F{n is exact only if U is independent of temperature. In the interest of accuracy it will often be advisable to fit the data with a second order polynomial in (7"abs — Tamb). For use in system calculations, on the other hand, the linearized efficiency equation will usually be much more convenient. In many applications the range of operating temperatures is sufficiently small that one obtains an acceptable approximation by linearizing the efficiency curve over the operating temperature range. Thus one replaces the efficiency curve by a straight line with intercept TJO equiv and slope — [/«,„„, both of which are calculated by a least squares fit over the operating temperature range in a specific application. This is illustrated in Fig. 4.1.2b and by Example 4.2.3 at the end of the next section. Cooper and Dunkle [1980] have examined this approach in full detail and provide an explicit numerical procedure for optimizing this linear fit. The error introduced by working with a constant equivalent U value at average operating temperature is likely to be smaller than errors due to unknown variations of weather conditions. On the other hand, the stagnation temperature, corresponding to zero efficiency of a collector, may be seriously overestimated if a low-temperature U value is assumed.
90
Active Solar Collectors and Their Applications
4.2 EFFICIENCY BASED ON FLUID TEMPERATURE In practice it is much easier to measure the temperature of the heat transfer fluid in the collector than the temperature of the absorber surface. Also, in practical applications one needs to know the efficiency as a function of the fluid temperature, not as a function of absorber temperature. Of particular interest are the fluid temperatures Tm and TOM at the inlet and outlet of the collector, respectively. Fortunately the efficiency equation is equally simple if it is based on fluid temperature rather than absorber temperature. As shown in Chapter 10, one can account for the fluid temperature by means of an additional multiplicative factor, the heat transfer factor. If the temperature rise along the collector is sufficiently small that the collector can be treated as if all the fluid were at the mean temperature
then the efficiency equation takes the form
where Fm is a collector parameter that accounts for the heat transfer from the absorber surface to the fluid. Fm is called collector efficiency factor or heat transfer factor and is also known by the symbol F. It depends on the construction of the collector but is practically independent of operating conditions. Typical values for Fm are in the range of 0.8-0.9 for nonevacuated air collectors, 0.9-0.95 for nonevacuated liquid collectors, and 0.95-1 for evacuated collectors. EXAMPLE 4.2.1
A flat plate collector of aperture area A = 2 m2 is tested for instantaneous efficiency. Water is pumped through the collector at a flow rate m = 0.03 kg/sec. A pyranometer mounted in the collector plane reads an insolation of 7 = 950 W/m2. Ambient air temperature is Ta = 10°C and the collector inlet and outlet fluid temperatures are Tm = 10°C and 7"out = 20°C. The test is repeated under the same ambient conditions but with higher inlet temperature, rin = 60°C, and yields TOM = 65°C. (a) What is the efficiency at Tin = 10°C? (b) What is the efficiency at Tm = 60°C? (c) What are the collector parameters Fmrj0, and FmU if the efficiency curve is linear? (d) How hot can this collector get when no heat is extracted? SOLUTION
Collector Efficiency and Collector Testing
91
for mean fluid temperature
Linear efficiency equation, based on Tm:
This system of two equations for two unknowns has the solution
(d) The stagnation temperature corresponding to peak insolation (/max 1000 W/m2) on the hottest days (ramb ^ 40°C) is
(obtained by setting ?j = 0). For some applications, it is more convenient to specify the fluid inlet temperature rin than Tm. In terms of T[n the equation is
where Fin is called the heat removal factor, denned by the equation
92
Active Solar Collectors and Their Applications
Fm and Fin are called heat transfer factors, and the equations are derived in Chapter 10. Equations (4.1.12), (4.2.2), and (4.2.3) can be summarized in the form
where the heat transfer factor F depends on the temperature base for the collector temperature Tcott. It is given by
To give a physical interpretation we note that the absorber surface temperature rabs must be higher than the fluid temperature Tm in order for heat to flow from the absorber surface to the fluid.4 If the fluid temperature is specified as Tcon then the corresponding absorber temperature is higher than Tco]]; this effect is reflected by the fact that the heat transfer factor Fm < 1. Similarly if the inlet temperature Tm is specified as TcM, the mean fluid temperature Tm is higher than TcM and the efficiency must be lower than if Tm were equal to TmU. Hence the heat transfer factor Fin must be less than Fm. F'm approaches Fm only in the limit of infinite flow rate because then the entire collector fluid remains at the inlet temperature. The stagnation temperature is the same for all three equations because inlet, outlet, and absorber temperature become equal when no heat is extracted. When measuring the efficiency as a function of(Tco[[ — Tamb)/I one determines (if one uses a linear fit for the data) two collector constants: the intercept (Fr]0) and the slope (FU). Using the fluid temperature as base, one does not measure F separately from r?0 and U. This presents no problem because in practice one will use only the equations that are based on T-m or Tm and one can convert test data from Tm to Tm or vice versa without knowing Fm by itself. For example, if test data have been reported in the form
one can convert them to the form
by writing
4 Black fluid collectors are an exception, in a sense, because they absorb the solar radiation directly in the fluid. The glazing that contains the black fluid can therefore be at lower temperature, and Fm can be greater than unity if it is defined relative to the surface of that glazing [Minardi and Chuang, 1975],
Collector Efficiency and Collector Testing
93
The ratio FJFm can be found by dividing both sides of Eq. (4.2.4) by Fm:
Nowhere in the last three equations does Fm appear by itself, and hence only the products (Fm^0) and (FmU) are needed to calculate Finjj0 and FinC7. To convert in the other direction one first solves Eq. (4.2.4) for FmU
and then writes Fmrj0 as
Again, it is clear that one need not know Fin by itself but only the products (-Fin?70) and (Fm U), and, of course, the flow rate used in the collector test. EXAMPLE 4.2.2
A flat plate collector has been found to have the following parameters: Fmrj0 = 0.726 and FinU = 3.62 W/m2°C. The collector area is A = 1.4 m2 and the flow rate and specific heat of the heat transfer fluid used during the test were m = 2.8 X 10~2 kg/sec and c = 4.18 X 103 J/kg°C. Convert these data to mean fluid temperature. SOLUTION
First find me/A:
then use Eq. (4.2.10)
94
Active Solar Collectors and Their Applications
There has been some debate over which temperature base is more useful, Tm or Tm. In principle, it does not matter for the linear efficiency equation because one can convert from one base to the other. In practice, however, Tm is preferable. First of all, one needs to specify only two parameters, Fmr]0 and FmU, whereas -Fin?;0 and FinU data are useless unless the flow rate (or rather me) during the test is stated in addition; me must be specified for the rin case because Fin varies strongly with flow rate. A further point in favor of Tm lies in the fact that Eqs. (4.2.3) and (4.2.4) are exact only if the efficiency equation is linear. A nonlinear fit based on rin is more difficult to convert to other flow rates. Many equations in the remainder of this book hold equally well regardless of which temperature base is chosen. We shall therefore write all these equations in the general form of Eq. (4.2.5), where the heat transfer factor F is understood to correspond to the temperature base according to Eq. (4.2.6). Originally the instantaneous efficiency equation, Eq. (4.2.5), had been derived for flat plate collectors. However, it is perfectly general and provides a convenient three-parameter description (optical efficiency, U value, and heat transfer factor) of all active solar collectors. In fact, it can even be applied to the salt-gradient solar pond, as shown by Kooi [1979]. EXAMPLE 4.2.3
Consider the solid line labeled "integrated CPC Tubes" in Fig. 4.4.4. (Its efficiency is higher than the measured data points, because the collector is projected to have lower heat losses from the end portions). The curvature of this line reflects the fact that the U value really increases with temperature. Approximate this curve by a straight line and evaluate the accuracy of this approximation by the following steps. (a) Obtain a three-parameter curve fit in the form
that agrees with the curve at AT1///, = 0, 0.1, and 0.2°C m2/\V, assuming that Ib = 850 W/m2. (b) How would the parameters Fmrj0, FmU0, and FmU\ change if the beam irradiance during the test had been only 800 W/m2? (c) Suppose this collector is to be used over a temperature range of Tm = 90°C to 170°C while Tamb = 10°C and Ib = 800 W/m2. Estimate graphically the parameters Fmrj0efiuiv and FmUeqniv of the equivalent linearized efficiency equation that best represents the actual curve under these conditions. (d) What are the errors at Tm = 0°C, 90°C, 130°C, and 170°C if one uses this linearized equation instead of the exact curve? SOLUTION
(a) In this problem all efficiencies are based on /,„ hence let us drop the subscript b. Reading Fig. 4.4.4. we find
Collector Efficiency and Collector Testing
95
These three points are to be fit by an expression
The AT = 0 point gives directly Fmr,0 = 0.71. Now the Ar/7 = 0.1 and 0.2 points yield two equations for two unknowns FmU0 and FmU{:
and
The solution is FmU0 = 0.4 W/m2 °C and FmU{ = 2.35 X 10~3 W/m2 °C2. Note that the U value doubles over the range from AT = 0 to 170°C, being FmU = 0.4 W/m2 °C at AT = 0 and FJJ = 0.8 W/m2 "CatAT = 170°C. (b) With 7 = 800 W/m 2 instead of 850 W/m2 we would have obtained the following results:
while the other two equations would become
and
This yields FmU0 = 0.4 W/m2 °C, the same as before, and FmU{ = 2.5 X 10~3 W/m2 °C2, slight change. At Tm = 170°C we would get FJJ = 0.4 + 0.425 = 0.825 W/m2. (c) Tm = 90°C, ramb = 10°C and 7 = 800 W/m2 correspond to AT/1 = 80°C/800 W/m2 = 0.1 °C m2/W. Tm = 170°C at ramb = 10°C and 7 = 800 W/m2 corresponds to AT/7 = 0.2 °C m2/W. Hence, draw a straight line that best approximates the efficiency curve over the range from AT/I = 0.1 to AT/I = 0.2. The result is
and
96
Active Solar Collectors and Their Applications
Both intercept and slope are much higher than for the real efficiency curve; in other words, this would be a bad approximation at high or low AT. But let us see how good it is over the range from 90°C to 170°C. (d) At Tm = 90°C, A 777 = O.TC m2/W and the efficiency is 77 = 0.65 (exact curve), while the linearized expression yields
The linearized fit is slightly high, but the error is comparable to the uncertainty in reading the efficiency curve.
4.3 INCIDENCE ANGLE MODIFIERS In the previous section we have assumed that the efficiency curve has been measured at normal incidence. However, in most collectors the optical efficiency changes with angle of incidence 9. For example, the optical efficiency of flat plate collectors decreases as the angle of incidence increases because of increased reflection losses from cover and absorber. In a parabolic trough there is, in addition, increased spillover of radiation. The incidence angle affects only the optical efficiency, and one accounts for it by writing
where K(0) is the incidence angle modifier. In this equation, 7 is still the insolation on the aperture; hence, K(6) does not include the reduction of insolation due to cos d. For some collectors, in particular, collectors with tubular absorbers, the incidence angle modifier K = K(6t, 8±) depends on two angles 8^ and 0j_, the projected incidence angles parallel and perpendicular to the tube direction. Even in that case, one may be able to simplify data taking and analysis by approximating AT(0y, 0j_) by a product of two separate incidence angle modifiers ^,(0,,) and KL(eL) [Mclntire, 1982]. For flat plate collectors K(8) can usually be approximated by a straight line when plotted against the variable I/cos 6 — 1 , and a single coefficient b0 suffices to characterize K(0):
Collector Efficiency and Collector Testing
97
This is shown in Fig. 4.3.1 for some typical flat plate collectors. This fit has no fundamental significance and can become very bad indeed for angles beyond 70°. In fact, at 90°, K(8) must vanish whereas Eq. (4.3.2) does not. For tubular collectors Eq. (4.3.2) is not recommended; this becomes obvious from Fig. 4.3.2, which shows K(ff) for several evacuated tubular collectors. For parabolic troughs it is advisable to separate K(9), which is inde-
Figure4.3.1 Incidence angle modifier for three flat plate collectors, (a) Plotted versus 6. (b) Plotted versus I/cos 6 — 1 (From Duffie and Beckman [1980]; reprinted with permission of John Wiley & Sons).
98
Active Solar Collectors and Their Applications
Figure 4.3.2 Incidence angle modifiers for several evacuated-tubular collectors, (incidence angle in direction perpendicular to tubes) (From Owens-Illinois [1979]). pendent of collector length, from an end loss factor r(0) by writing 77 as
T(0) accounts for the spilling of radiation over the end of a finite trough. If the receiver tube has the same length as the reflector trough and is placed symmetrically then T(0) is given by
Figure 4.3.3 Incidence angle modifier for several parabolic troughs, versus I/cos 6 — 1 (From Gaul and Rabl[ 1980]).
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where / = trough length = receiver length w = trough width / = focal length The incidence angle modifier K(6) for several commercial parabolic troughs is plotted in Fig. 4.3.3; one can see that for most of the troughs a straight line versus the variable I/cos 0 — 1 does not yield a very good fit. Strictly speaking, one should monitor diffuse and direct radiation separately when measuring K(6) for nontracking collectors, because direct and diffuse radiation have different incidence angles. In practice, one usually does not worry about this detail since collector tests are done on clear days when the contribution of diffuse radiation is small.
4.4
TEST RESULTS
Test results for some typical collectors are presented in Figs. 4.4.1-4.4.7. They are plotted as efficiency versus (Tmll — 7"amb)/7. Temperature base (i.e., inlet temperature 7"in or mean fluid temperature Tm) and insolation base (beam irradiance Ib or hemispherical irradiance 7A) are indicated in each figure. If the incidence angle modifier has been measured, it is also shown, as a graph or table. The reader is warned that efficiency data have uncertainties on the order of 5% or more, unless extreme care was taken in designing the test equipment and in carrying out the tests. This is illustrated by the scatter of data points in Fig. 4.5.5. A recent and comprehensive summary of flat plate test results can be found in Kirkpatrick [1983]; this reference also discusses the accuracy of test data. The instantaneous efficiency curve, together with the incidence angle modifier, is needed for predicting the performance of solar energy systems
Figure 4.4.1 Instantaneous efficiency relative to Ih for three flat plate liquid collectors (1,2, and 3), a flat plate air collector (4), and an evacuated tubular collector (5) (Based on Hill et ai. [1979] and on Owens-Illinois [1979]).
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Figure 4.4.2 Instantaneous efficiency (relative to IH) and incidence angle modifier of new Sunmaster evacuated tubular collector with 1.1X CPC reflector (Based on Sunmaster brochure, Corning, N.Y. [ 1983]).
and for selecting the most cost-effective collector. In general one should not select a collector on the basis of the instantaneous efficiency curve alone. Rather one should use as selection criterion the ratio of useful delivered energy and system cost. This depends not only on the collector but also on system design and operating conditions as discussed in Chapters 11 and 12.
Figure 4.4.3 Summary of the performance curves for several nonevacuated CPC collectors relative to Ih. The 5.2X and 3X curves are referenced to standard very clear day conditions. Typical double-glazed and singleglazed flat plate performance is shown for comparison (Adapted from Rabl et al. [1980]).
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Figure 4.4.4 Instantaneous efficiency curves for several collectors, plotted versus beam irradiance /b. "Integrated CPC Tubes" refers to nontracking fixedtilt collector with evacuated glass tubes shaped as 1.5X CPC. Data are measured with 0.94-m long prototype; curve is projection for 1.83-m long commercial module (From O'Gallagher et al. [1982]).
4.5 4.5.1
STANDARD COLLECTOR TEST PROCEDURES Instantaneous
efficiency
Several standard collector test procedures have been published [ASHRAE, 1977; BSE, 1978; Hill, 1976]. They differ in details such as the choice of the temperature base. The ASHRAE 93-77 standard reports efficiency data relative to collector fluid inlet temperature, while the BSE standard and Hill [1976] specify the mean fluid temperature as base. The latter is more convenient for applications where the flow rate differs from the test conditions.
Figure 4.4.5 Efficiency relative to /b for the Solar Kinetics T-700 Solar Collector with glass reflector surface (From Harrison [1981]).
Figure 4.4.6 GE parabolic dish collector efficiency relative to /„ vs delta temperature/insolation (From Dudley and Workhoven [1979]).
Figure 4.4.7 Efficiency relative to 7b and incidence angle modifier of advanced parabolic trough (Adapted from Sandia [1980]).
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Other differences include specifications for preconditioning and survival testing under stagnation and the use of gross versus net area (ASHRAE 9377 is based on gross area). Jenkins and Hill [1980] have compared the ASHRAE and the BSE test procedures and found that the BSE procedure yields more accurate and repeatable results while requiring less testing time. The basic ingredient of these test procedures is the same. A heat transfer fluid is pumped through the collector, and the delivered heat Q [in W]
is determined by measuring flow rate ra [in kg/sec], specific heat c[in kJ/kg K], and inlet and outlet temperatures Tm and TOM. Care must be taken to ensure good mixing of the fluid in the vicinity of the temperature probe; otherwise the difference between the wall and bulk fluid temperature may cause erroneous measurements. Special mixing tubes are commercially available that can be inserted just upstream of the temperature probe. Adequate mixing can usually be achieved by inserting spoilers such as copper wool or spirals in the flow channel if the flow is not turbulent.5 The most reliable results are obtained by using water as the heat transfer fluid and by measuring flow rates with a bucket and stopwatch. Since many concentrating collectors are expected to operate above 100°C, however, other heat transfer fluids will have to be considered for operation at atmospheric pressure: water plus ethylene glycol (good to about 120°C) ethylene glycol (good to about 120°C) heat transfer oils such as Dowtherm, Therminol, or silicon oil (good for the range 100-350°C) air (temperature range unlimited) It is difficult to get reliable data for flow rate and specific heat, and it is advisable not to trust flow meters and published values of heat capacity without independent verification. Mixtures, for example, water plus antifreeze, are not recommended because selective evaporation may change the composition of the fluid during the course of a test series. A simpler and more reliable alternative is the use of a calibrated heat source as shown in Fig. 4.5.1 [Reed, 1977; Collares-Pereira et al., 1981]. An electric resistance heater is used to preheat the heat transfer fluid before it enters the solar collector. (Placement downstream from the collector may result in excessive fluid temperatures.) If the electric heat source is well insulated against thermal loss to the environment, all the electric input (2ei = voltage X current 5 It is advisable to check that the pumping energy dissipated in the collector is negligible compared to the solar input; otherwise a correction must be applied.
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Figure 4.5.1 Calibrated heat source for measuring collector efficiency.
goes into raising the fluid temperature by an amount
Determination of collector output
is thus reduced to a measurement of four temperatures and of electric power, which can be accomplished with ease and accuracy. (If alternating current is to be used in locations with significant phase angle between current and voltage, use of an ac power meter is recommended.) This method eliminates the need to know either flow rate or heat capacity. A crucial requirement for any direct measurement of instantaneous efficiency is the need for steady state conditions. Obviously insolation and flow rate must be constant. In addition, both Tm and rout must be constant. Constancy of the temperature rise (!Tom — Tm) across the collector is not sufficient because of the finite transit time of the fluid through the collector. Since T-m and TOM are measured simultaneously they correspond to different fluid elements; thus the collector could be warming up or cooling down even though (rout — rin) is constant. The importance of this point and the difficulty of obtaining adequate steady state conditions are illustrated by Fig. 4.5.2, which purports to show a purely optical quantity, the incidence angle
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Figure 4.5.2 Data on incidence angle modifier for a double-glazed water heating collector, illustrating the effect of collector warm-up and cool-down (Adapted from Hill etal. [1979]).
modifier. Even though the test prescriptions for constancy of test variables were followed, the measured efficiency points are significantly higher in the afternoon when the collector is cooling down than in the morning when the collector is warming up. In other words, the test data of Fig. 4.5.2 do not represent true steady state conditions. Several steps can be taken to minimize the effects of transients on collector testing. When the transient effects are small, one obtains a good approximation to the true answer by averaging results during warm up and during cool down. To maximize the length of time of nearly constant sunshine, one should mount the collector on a moveable platform in order to keep the incidence angle of the sun on the collector small. The platform need not track the sun continuously; it is sufficient to adjust the direction periodically, say every half hour. If the collector is light enough, one does not need a rotating platform and one can readjust the collector by hand. The ability to keep the collector nearly normal to the sun is well worth the extra effort; it greatly speeds up the testing process and it improves the accuracy. In fact some high temperature collector test loops have such long warm-up times that it is nearly impossible to get reliable data without moveable collector platform. Transient effects arise not only from the heat capacity of the collector. More important is the heat capacity of the test loop itself, if the loop is
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Active Solar Collectors and Their Applications
closed as shown in Fig. 4.5.3. The storage tank serves to smooth transients due to small fluctuations, but it is unlikely to keep the temperatures constant during morning warm-up, unless the controls are excellent. A far simpler method for eliminating transient effects relies on an open test loop, as shown in Fig. 4.5.4. The temperature of ambient air or water from the water mains is sufficiently constant to guarantee constant inlet temperature to the collector. The use of an electric heater on the collector inlet side permits the attainment of any desired inlet temperature. Of course, one can afford to run an open loop only when the fluid is sufficiently cheap; in practice this restricts open loop operation to water and air. If water or air are acceptable as test fluids, the use of open test loops is recommended as the most convenient and accurate method by far. Finally, as an illustration of the errors to be expected in standard collector efficiency tests, we show in Fig. 4.5.5 the test results for the same collector but carried out by six different laboratories. The scatter of the individual laboratories about the average can easily be as large as ± 5% even though all followed the same test procedure. 4.5.2
Collector time constant
The performance of a collector under intermittent sunshine depends on its heat capacity. The higher the heat capacity, the longer the collector takes to warm up and cool down. During warm-up the efficiency is lower, during
Figure 4.5.3 Closed-loop test setup for liquid heating solar collectors (From Hill etal. [1979]).
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Figure 4.5.4 Open-loop through-flow test setup for liquid heating solar collectors (From Hill etal. [1979]).
cool down higher than in steady state. If all heat above ambient temperature T"amb can be utilized, then any heat stored in the collector can be extracted and there is no net loss due to transients. This is the case in certain industrial hot water and hot air systems where the collector is employed as preheater only (See Section 12.3). However, if heat is useful only above a certain threshold higher than T"amb, then transients do impose a penalty. An accurate analysis of collector transients is quite complicated. To see what considerations are involved, let us assume that the absorber can be characterized as a single thermal node of mass Mcoll and specific heat cmn, at a uniform temperature equal to the fluid temperature Tm. This is an approx-
Figure 4.5.5 Instantaneous efficiency data for the same flat plate collector as measured by six different laboratories, all using the ASHRAE 93-77 test procedure (Adapted from Streedetal. [1979]).
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Active Solar Collectors and Their Applications
imation because really different parts of the collector and of the fluid are at different temperatures. Let us further assume that the transit time of the fluid through the collector is short compared to the cool-down or warm-up time. Then the transient energy balance of the collector is given by
where the last term represents the heat extracted by the heat transfer fluid. Inserting the fluid temperature Tm = (r in + Toni)/2, one can solve this differential equation for TOM once the time dependence of /, ramb, and T-m are specified. The warm-up rate depends on insolation, inlet temperature, heat loss, and flow rate in addition to collector heat capacity. In particular the warm-up rate is different from the cool-down rate. The ASHRAE test procedure attempts to determine the transient collector behavior by measuring the cool-down rate when the collector is suddenly and completely shaded while flow rate and inlet temperature remain constant with rin = ramb. In that case Eq. (4.5.3) reduces to
This equation has the solution
with the time constant
An example of a time-temperature plot for the time constant is shown in Fig. 4.5.6. The time constant is the time at which TOM — ramb has decreased to l/e of its value at t = 0. 4.6 SIMPLIFIED COLLECTOR TEST PROCEDURES Accurate (i.e., with less than 10% error) measurement of efficiency under actual flow conditions is a challenging task, especially at high temperatures. The cost and effort involved in building a high-temperature collector test station can be justified only if many collectors are to be tested; otherwise the services of established test facilities should be called upon. However, the performance of a solar collector can be determined indirectly by a number of tests that are relatively simple. These tests, which are described in the following subsections, can be carried out on small collector
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Figure 4.5.6 A time-temperature plot for a flat plate air heater showing temperature drop on sudden interruption of the solar radiation on the collector. The time constant is the time for the temperature to drop to Ije of the total potential drop, that is, for B/A to reach 0.368 (Adapted from Hill et al. [1979]).
modules; thus they are a valuable tool that permits evaluation of a preliminary design without the need to construct a complete (and expensive) collector prototype. These tests are recommended whenever a new collector is to be designed. All the important collector parameters, in particular, the U value, the optical efficiency, and the incidence angle modifier, can be determined by these simplified tests. Only the heat transfer factor Fm is not obtained by these tests. 4.6.1 Measurement of heat loss The U value can be measured most easily by heating the absorber electrically either in the laboratory or outdoors at night. For this purpose, heating wire, for example, Nichrome, is inserted into the absorber flow channel, and electric power (2ei and the resulting equilibrium absorber temperature rabs are measured. The U value, relative to collector aperture area A, is then
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Active Solar Collectors and Their Applications
The heat loss of a real collector may differ slightly from this value. For example, the manifolding (plumbing connections) may be different. In collectors with reflectors adjacent to the absorber, further differences arise from the "superefficiency effect" due to hot reflectors, as discussed in Section 9.10. Usually the differences are sufficiently small not to detract from the value of the information gained by this test. 4,6.2
Warm-up test
For this test one allows the collector to warm up under full sunshine. If the mass of the absorber is M and the specific heat c, then the temperature of the absorber increases according to the energy balance equation
When the absorber is at ambient, Tabs = ramb, Eq. (4.6.2) gives the optical efficiency as
As the absorber warms up, heat losses become more important. This requires that U be known. U can be measured by turning a collector with hot absorber away from the sun. For nonconcentrating collectors this means taking the collector indoors to eliminate any contribution of diffuse radiation. With 7 = 0 , Eq. (4.6.2) yields the U value from the cool down rate as
Having measured U, we can correct Eq. (4.6.3) for heat loss effects during warm up by solving the differential equation (4.6.2). 4.6.3 Masked stagnation test The equilibrium temperature that an absorber approaches when no heat is extracted from the collector is called stagnation temperature rstag. Setting the left hand side of the efficiency equation (4.6.2) equal to zero, one obtains the following relation between collector parameters, insolation, and stagnation temperature:
Collector Efficiency and Collector Testing
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The U value increases with temperature, and in practice one wants to know the U value at the anticipated operating temperature. Therefore one should reduce the insolation / for this test to ensure that rstag is in the operating temperature range. The insolation can be reduced by placing a mask of known transmittance (e.g., a perforated sheet) in front of the collector aperture. The perforations must be small relative to the size of the receiver. The backside of the mask must be black to avoid multiple reflections between the mask and the collector. The mask should not interfere with the heat transfer from the collector to ambient. Measurements obtained by this procedure are described in Rabl et al. [1980] and Gordon et al. [1980]. The warm-up test and the masked stagnation test can be performed at any incidence angle. By repeating these tests at different incidence angles one can thus determine the incidence angle modifier.
4.7 MEASURING THE OPTICAL QUALITY OF FOCUSING COLLECTORS Evaluation of the optical quality of a solar concentrator is important: to the designer, to tell whether a collector needs improvement; and to the manufacturer, to ensure proper quality control. In addition to the optical efficiency 7/0, it is important to have a measure of optical errors and of losses due to reflected radiation missing the receiver. Some methods for measuring the contour accuracy of solar concentrators require either laser ray tracing or flux mapping at the receiver surface. Both approaches can provide accurate results, but the equipment is specialized and expensive and demands a good deal of time and/or expertise. However, one can also determine the optical errors by using the instantaneous efficiency measurements that are performed as part of a standardized performance evaluation. This can be accomplished by misaligning the collector slightly away from the sun and measuring the efficiency for several values of the misalignment angle. The optical error is then extracted by finding the theoretical curve that best fits these misalignment data. In a sense, this method employs the receiver itself as flux mapper. Bendt et al. [1980] describe this method in detail and present test results. As a much simpler qualitative test an optical inspection can be valuable. For this purpose one simply looks at the collector aperture and estimates what fraction of the aperture appears black [Wood, 1981]. Just one note of caution must be heeded: The distance from the eye to the collector aperture must be large compared to the width of the collector to ensure that all rays from the eye to the collector are sufficiently parallel compared to the acceptance angle to be measured. This distance depends on the collector. For a CPC with an acceptance angle of 60° and an aperture width 0.20 m a distance of several meters will be adequate, whereas for a high concentration parabolic dish with a 10-m diameter several kilometers and a telescope will be necessary. This inspection test can be made quantitative if one uses a camera with den-
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sitometric film. It has been successfully demonstrated by Masterson and Gaul [1981]. REFERENCES ASHRAE. 1977. "Collector Test Procedure of the American Society of Heating, Refrigeration and Air Conditioning Engineers." Standard 93-77. New York: ASHRAE. BSE. 1978. "BSE Guidelines and Directions for Determining the Utilizability of Solar Collectors, A Solar Collector Efficiency Test." Bundesverband fuer Solarenergie, Kruppstrasse 5, 4300 Essen 1, Federal Republic of Germany. Collares-Pereira, M. et al. 1981. "A Calorimeter for Solar Thermal Collector Testing." Solar Energy 27:581. Cooper, P. I. and Dunkle, R. V. 1981, "A Nonlinear Flat-Plate Collector Model." Solar Energy 26:133. Dudley, V. E. and Workhoven, R. M. 1979. "Summary Report: Concentrating Solar Collector Test Results Collector Module Test Facility January—December 1978." Report SAND 78-0977. Albuquerque, NM: Sandia National Laboratories. Gaul, H. W. and Rabl, A. 1980. "Incidence Angle Modifier and Average Optical Efficiency of Parabolic Trough Collectors." Trans ASME, J. Solar Energy Eng. 102:16. Gordon, J. M., Govaer, D., and Zarmi, Y. 1980. "Temperature-dependent Collector Properties From Stagnation Measurements." Solar Energy 25:465. Harrison, T. D. 1981. "Midtemperature Solar Systems Test Facility. Predictions for Thermal Performance Based on Test Data." Report SAND 80-1964/7. Albuquerque, NM: Sandia National Laboratories. Hill, J. E. 1980. "Standard Procedures for Collector Performance Testing." In Chapter 15, Solar Energy Techology Handbook, Dickinson, W. C. and Cheremisinoff, P. N., editors. New York: Dekker. Hill, J. E., Jenkins, J. P., and Jones, D. E. 1979. "Experimental Verification of a Standard Test Procedure for Solar Collectors." National Bureau of Standards NBS Building Series 117. Washington, D.C.: U.S. Dept. of Commerce. Jenkins, J. P. and Hill, J. E. 1980. "A Comparison of Test Results for Rat-Plate Water-Heating Solar Collectors Using the BSE and ASHRAE Procedures." ASME J. Solar Energy Eng. 102:2. Kirkpatrick, D. L. 1983. "Flat Plate Solar Collector Performance Data Base and User's Manual." Report SERI/STR-254-1515. Golden, CO: Solar Energy Research Institute. Kooi, C.F. 1979. "The Steady State Salt Gradient Solar Pond." Solar Energy 23:37. Lunde, P.J. 1980. Solar Thermal Engineering. New York: John Wiley & Sons. Masterson, K. and Gaul, H.W. 1981. "Optical Characterization Method for Concentrating Solar Collectors Using Reverse Illumination." SERI Report TP-641-1179. Golden Co. Solar Energy Research Institute. Mclntire, W.R. 1982. "Factored Approximations for Biaxial Incident Angle Modifiers." Solar Energy 29:315. Minardi, J.E. and Chuang, H.H. 1975. "Performance of a Black Liquid Flat Plate Solar Collector." Solar Energy 17:179. O'Gallagher, J.J., Snail, K., Winston, R., Peek, C., and Garrison J.D. 1982. "A New Evacuated CPC Collector Tube." Solar Energy 29:575.
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Owens-Illinois, 1979. "A Pipeline to the Sun From Owens-Illinois." Brochure SUNPAK No. 13.11/OW. Owens-Illinois, Inc. Development Center, 1020 N. Westwood, Toledo, OH 43607. Rabl, A., O'Gallagher, J., and Winston, R. 1980. "Design and Test of Nonevacuated Solar Collectors with Compound Parabolic Concentrators." Solar Energy 25:335. Reed, K. A. 1977. "Test Plan for Low Concentration Thermal Collectors." Informal report. Argonne National Laboratory. SANDIA. 1980. "Line Focus Solar Thermal Energy Technology Development. FY 1979 Annual Report." Report SAND 80-0865. Albuquerque, NM: Sandia National Laboratories. Streed, E. R., Hill, J. E., Thomas, W. C., Dawson, A. G., and Wood, B. D. 1979. "Results and Analysis of a Round Robin Test Program for Liquid Heating Flat Plate Solar Collectors." Solar Energy 22:235. Tabor, H. 1978. "Testing of Solar Collectors." Solar Energy 20:293. Tabor, H. 1980. "Letter to the Editor." Solar Energy 24:113. Wood, R. L. 1981. "Distant Observer Techniques for Verification of Solar Concentrator Optical Geometry." Report UCRL-53220. Livermore, CA: Lawrence Livermore Laboratory.
5. FUNDAMENTALS OF OPTICS FOR SOLAR COLLECTORS
The optical analysis of a solar collector has as its goal the calculation of the amount of radiation that is absorbed by the absorber. How complicated this calculation is depends on the collector type and on the desired accuracy. For example, in a single glazed flat plate collector with cover transmittance r and absorber plate absorptance a the fraction of the incident radiation that is absorbed by the absorber can usually be approximated by ra. If one wants to be more exact, one may have to include multiple reflections with attention to incidence angles and polarization. If T and a depend on wavelength, one may have to carry out separate calculations for different portions of the solar spectrum. In concentrating collectors one may have to calculate ray diagrams, and the calculations can become quite complicated. Needless to say, we will not analyze every collector type to the last detail. Rather we will develop the tools necessary for such an analysis and illustrate them by typical examples. We will not explicitly indicate any spectral dependence in material properties, and in most applications a calculation with properties averaged over the solar spectrum will be adequate. Only when two or more materials in one collector display strong spectral variation will a breakdown into separate spectral regions be necessary. However, even in that case, the same formulas apply, only with different parameters for different spectral regions. In order to understand solar collectors one does not need to study the entire subject of optics.1 The purpose of a collector is to collect radiation; preserving a faithful image of the sun is irrelevant. Accordingly this chapter presents only those aspects of optics that are most relevant for solar energy; namely, the optics of radiative heat transfer. Section 5.1 deals with reflection. Section 5.2 treats refraction and the transmission of radiation through collector covers. Section 5.3 discusses fundamental limitations of radiation concentrators. Section 5.4 presents a statistical analysis of image spread due to extended sources and optical errors. Section 5.5 shows how the width of the sun must be corrected if line-focus collectors are analyzed in a simple two-dimensional framework. Further topics which are needed only for specific collector types will be addressed in Chapters 6-8. 'For a classic text on optics the reader is referred to Born and Wolf [1975].
,14
Fundamentals of Optics for Solar Collectors 5.1
5,1.1
115
REFLECTION
The law of reflection
For ray tracing in three dimensions the use of vector notation is convenient. In terms of the three unit vectors in Fig. 5.1.1, i = direction of incident ray, n = direction of normal of reflector surface, f = direction of reflected ray, all three pointing away from the surface, the law of specular reflection states that (1) the angle of incidence equals the angle of reflection:
and (2) i, n, and f lie in the same plane:
Given any two of these vectors, the third one is uniquely determined by Eqs. (5.1.1) and (5.1.2), apart from trivial minus signs. If i and r are specified, n must be a linear combination ai + br because of coplanarity, and the coefficients a and b are fixed by Eq. (5.1.1) and by normalization. The result is
On the other hand, with i and n given, a similar argument shows that r is
Figure 5.1.1 Unit vectors for the law of specular reflection: i = incident ray, n = reflector normal, f = reflected ray.
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Active Solar Collectors and Their Applications
The fraction of the radiant energy that is reflected is given by the reflectivity p of the surface. If a beam undergoes n reflections at surfaces with reflectivity p on its way from aperture to absorber, then it is attenuated by a factor -r = p". In nonimaging concentrators, e.g., the CPC, and in radiation passages, e.g., light tubes, the number of reflections is different for different rays with different angles and points of incidence. Even in this case the fraction r of the incident radiation that is transmitted can be calculated with excellent accuracy by the formula
if « is taken as the average number of reflections. For a large class of concentrators and radiation passages the average number of reflections can be calculated in closed form [Rabl, 1977a]. 5.7.2
Optics oftroughlike reflectors—projection of ray trace diagram
For troughlike reflectors a two-dimensional analysis is sufficient, even at nonnormal incidence. Suppose that the trough is placed parallel to the z axis. The projections 0^xy and 6T<xy of the angles of incidence 6-l and reflection 6r on the xy plane, shown in Fig. 5.1.2, are given by
and similarly
Figure 5.1.2 Projections 6Ln. and 0,_V1. of angles of incidence 6, and of reflection 8,. on the xy plane; i.e., the plane perpendicular to the trough direction. The projected angles 8,n. and 8rn, are equal no matter how large the elevation of the incident ray from the plane of the paper.
Fundamentals of Optics for Solar Collectors
117
If nz = 0 then rz = —iz by Eq. (5.1.4). Furthermore, i • n = r • n by Eq. (5.1.1); hence it follows that
Thus, in any troughlike reflector aligned along the z axis, all incident rays with the same xy projection (plane of the paper in Fig. 5.1.2) are represented by the same two-dimensional ray trace diagram, no matter how large their elevation from the xy plane. Therefore rays with the same x, y components but different z components need not be traced separately. If a planar ray entering in the direction i = (ix, iy, 0) has been found to leave in the direction f = (rx, rv, 0), then a ray entering with
will leave with
no matter how many reflections have occurred. Since the ray trace diagram in troughlike reflectors is independent of the elevation of the incident ray from the xy plane, the focal length of a parabolic trough does not change with elevation. This is not true for two-dimensional refractive concentrators; the focal length of a linear Fresnel lens does change with elevation. There are, however, two properties of troughlike reflectors for which the elevation of the sun from the projection plane does make a difference. First, there is the end effect of finite troughs; this end loss can be calculated by Eq. (4.3.4) (unless it is eliminated by adding flat end reflectors). Second, there is an increase in the projected angular width of the sun which necessitates a larger absorber, as discussed in Section 5.5. 5.2 5.2.7
REFRACTION Snell's law of refraction
Snell's law of refraction states that the incident ray i, the normal to the surface n, and the refracted (transmitted) ray t lie in a plane and that the angle of incidence 6, = arccos (i • n) and the angle of refraction 6, = arccos (t • n) satisfy
where «, and n, are the indices of refraction of the two media. In vector notation coplanarity of i, n, and t implies that t must be a linear combina-
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Active Solar Collectors and Their Applications
tion of i and n:
The coefficients a and b are uniquely determined (apart from trivial minus signs) by Eq. (5.2.1) and by normalization. One can readily verify that for the vectors indicated in Fig. 5.2.1 the refracted ray t is related to i, n, and refractive indices by
5.2.2
Fresnel equations
In addition to the direction of a refracted ray one needs to know how much radiation is reflected and how much is transmitted. The reflection coefficients r (equals ratio of reflected radiation over incident radiation) are different for the parallel and perpendicular components of polarization. Parallel and perpendicular refer to the plane spanned by the incident direction and the surface normal, as indicated by the subscripts , and j_. For radiation passing from a medium with refractive index «,• to a medium with index n, the reflection coefficients are given by
for the parallel component,
Figure 5.2.1 Unit vector for law of refraction: i = incident ray, n = surface normal, f = transmitted ray.
Fundamentals of Optics for Solar Collectors
119
for the perpendicular component, and
for unpolarized radiation. #, and 6, are the angles of incidence and refraction, related by Snell's law, Eq. (5.2.1). For radiation at normal incidence 0, and 6, vanish and Eq. (5.2.6) yields, in the limit 0, -* 0, the reflection coefficient
where
is the relative index of refraction. Most glazing materials have an index near 1.5 (See Chapter 13), and the corresponding reflection coefficient at normal incidence is
At grazing incidence, r approaches 1 . For intermediate angles rL increases monotonically as 0, increases, while r( first decreases until it vanishes at the Brewster angle where 0, + 9, = vr/2 and then increases to unity at 0, = ir/2. Direct solar radiation is unpolarized, and the partial polarization of diffuse sky radiation has negligible effects. Hence, one can consider incident solar radiation to the unpolarized. Nonetheless there could be polarization effects when calculating transmission of solar radiation through multiple covers. The difference between reflection and transmission coefficients for different polarizations implies that at nonzero incidence angle the transmitted radiation is partially polarized even if the incident radiation was not.
5.2.3 Multiple reflections In many applications radiation is to be transmitted through a slab of material; in that case there are two parallel interfaces, each with reflection losses. As a first approximation for small reflection coefficients r (at each surface) one can approximate the transmission through the slab by
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if there is no absorption in the slab. A more accurate calculation takes into account all the multiple reflections as shown in Fig. 5.2.2 for a nonabsorbing slab. The calculation is done separately for each component of polarization, keeping track of the fact that r]} and rL are functions of 0,. At the first interface a fraction (1 — rL) of the perpendicular component is transmitted into the slab. At the second interface (1 — rL}2 is transmitted, while (1 — rL)rL is reflected back up. Of the latter portion some will be reflected back down at the first interface, thus repeating the pattern. Contributions from successive round trips correspond to successive terms in a geometric series, and the sum of all transmitted terms is
The same formula holds for the other polarization component. Hence, the transmittance of unpolarized radiation is given by the average transmittance for the two components:
For the above example, a medium with n = 1.5 at normal incidence, we find
only slightly larger than the naive result 0.920 of Eq. (5.2.10). Of course the effect of multiple reflections becomes more important if the reflection coef-
Figure 5.2.2 Transmission of radiation through nonabsorbing slab of material (From Duffie and Bcckman [1980]; reprinted with permission of John Wiley & Sons).
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ficients are larger (as they indeed are at nonnormal incidence), or if there is more than one cover.2 For a system with TV identical covers, Duffie and Beckman [1980] give the result
again for the case where nothing is absorbed in the material. This equation has been plotted in Fig. 5.2.3 versus incidence angle for one, two, three, and four covers of glass with n = 1.526. One sees that the transmittance remains almost constant to 40°, then begins to drop, slowly at first, rapidly beyond 60°. At 90° the transmittance vanishes.
5.2.4 Absorption in the cover Absorption of radiation in a partially transparent medium is described by Bouguer's law. The fraction a of radiation that is absorbed over a path length L in a medium of extinction coefficient K is given by
For a slab of thickness d the path length is given by
2
A convenient method for analyzing a system of several covers has been presented by Edwards [1977].
Figure 5.2.3 Transmittance of radiation through 1, 2, 3, and 4 nonabsorbing covers with index of refraction 1.526 (From Duffie and Beckman [1980]; reprinted with permission of John Wiley & Sons).
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where 6, is the angle between the surface normal and the direction of the light ray in the slab. Bouguer's law holds for each wavelength X, but if K varies with X, one must replace Eq. (5.2.15) by a superposition of exponentials, as (for example) for solar pond calculations. For glass it is safe to assume a single extinction coefficient, which ranges from 4 rrr1 for "water white" glass to approximately 32 m~' for ordinary window glass (which appears greenish at the edge). Thus a d = 5 mm thick pane of ordinary window glass absorbs a = 1 — exp (—32 X 0.005) = 15% of the solar radiation at normal incidence. It is straightforward to modify Fig. 5.2.2 and the transmission Eqs. (5.2.11) and (5.2.12) for absorption in the slab. The result is
for the parallel component of polarization, with a given by Eq. (5.2.15). The analogous formula holds for the other polarization component, and the transmission coefficient for unpolarized radiation is the average of rt and T]_. For good materials the approximation
will be adequate. Numerical results for the transmission of cover systems with absorption are shown in Fig. 5.2.4 for a slab with refractive index 1.526 and various absorption coefficients as indicated by the values of Kd. The total reflectance p y of a slab with absorption is obtained in a similar manner
with the analogous formula for the p±. Note that p includes multiple reflections between the slab surfaces whereas r is the reflection coefficient for a single surface. 5.2.5 Absorption at the absorber Ultimately the solar radiation is, of course, to be absorbed by the absorber. Since no absorber is perfectly black, one has to consider multiple reflections between cover system and absorber plate, if one wants to be exact. This is shown in Fig. 5.2.5. The summation of multiple reflections leads again to a geometric series for the fraction of incident solar radiation absorbed by the absorber. This fraction is frequently called effective ra product and designated by (TO) (= optical efficiency ?j0 of flat plate):
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Figure 5.2.4 Transmittance of radiation through 1, 2, 3, and 4 covers with index of refraction 1.526 and various absorption coefficients (From Duffie and Beckman [1980]; reprinted with permission of John Wiley & Sons).
where a is now the absorptance of the absorber surface. For the cover reflectance we take pd, the diffuse reflectance, because most solar absorbers are diffuse reflectors. As shown by Duffie and Beckman [1980], pd can be approximated by evaluating Eq. (5.2.19) at an angle of incidence 6( « 60°. In practice the difference between (TO) and the simple product of r and a is likely to be very small. Even for selective coatings a will usually be at least 0.9 while the cover reflectance pd may be on the order of 0.2. For such a case the difference between (TO) and ra is 2%.
Figure 5.2.5 Effect of multiple reflections between absorber and cover.
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Active Solar Collectors and Their Applications
5.2.6 Optics of glass tubes In order to reduce heat losses in solar thermal collectors with tubular receivers, it may be desirable to place the absorber tube inside a glass tube. The effects of refraction in the glass tube on the optics of a solar concentrator can be determined by a simple method that follows from the connection between rotational symmetry and angular momentum conservation as shown by Rabl [1977b]. This is best expressed in terms of the impact parameter of a light ray, defined as the shortest distance between the light ray and the symmetry axis of the tube. In Fig. 5.2.6 the impact parameters r0, r,, and r2 of the light ray R outside the tube, in the tube wall, and inside the tube, respectively, are related by
if R is in the plane of the paper, and by
if R is nonplanar with z component sz. If R is tangent to the absorber tube in the absence of the glass envelope, it will also be tangent to this absorber (at a different point) after passing through the envelope. Therefore, the addition of a concentric glass tube does not alter the fraction of rays that is accepted by the absorber tube. These statements are true both for planar and for nonplanar rays, and they are independent of diameter and refractive index. From this one can also conclude that the effect of multiple reflections in the tube wall and between glass tube and concentric absorber tube can be calculated with the same formulas as for the flat plate case. This follows, because if a ray is in the right direction to hit the absorber a first time then all its associated multiple reflections will also hit the absorber.
Figure 5.2.6 Refraction of light ray by glass tube.
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5.3 CONCENTRATION OF SOLAR RADIATION
5.3.1 Definition of concentration ratio Concentration of solar radiation becomes necessary when higher temperatures are desired than can be obtained with a flat plate collector or when, as in the case of expensive receivers with evacuated envelopes, the cost of the receiver per unit area is higher than the cost of mirrors per unit area. Two definitions of concentration are natural and have been in use; to avoid confusion a subscript should be added whenever the context does not clearly specify which definition is meant. The first definition is strictly geometric, as ratio of aperture area to absorber surface area,
and the name geometric concentration is recommended. The second definition, in terms of intensity ratio at the aperture and at the absorber,
involves absorption effects in addition to geometry, and should be referred to as flux concentration. Although flux concentration is a useful concept in photovoltaic work, the geometric definition is more appropriate for solar thermal collectors; therefore throughout this book, "concentration" shall mean "geometric concentration," unless otherwise indicated. Closely related to the concentration is the acceptance angle 26a; i.e., the angular range over which all or almost all rays are accepted without moving all or part of the collector. As shown in the following section, the second law of thermodynamics requires that the maximum possible concentration for a given acceptance half-angle Ba is
for two-dimensional (troughlike) concentrators, and
for three-dimensional ones (cones, dishes, pyramids). Since the angular radius of the sun is A5 = 4.7 mrad the thermodynamic limit of a tracking solar concentrator is 213 in two-dimensional (line-focus) geometry, and 45,300 in three-dimensional (point-focus) geometry. In prac-
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tical systems one must increase the acceptance angle and hence reduce the concentration ratio for a number of reasons: most conventional concentrators, in particular, focusing parabolas or lenses, are based on optical designs that fall short of the thermodynamic limit by a factor of 2-4 (See Section 7.1) tracking errors and errors in mirror contour and receiver alignment necessitate design acceptance angles considerably larger than the angular width of the sun no lens or mirror material is perfectly specular; therefore the acceptance angle must be enlarged further because of atmospheric scattering, a significant portion pf the solar radiation may come from directions other than the solar disk itself The choice of optimal concentration for a given application involves evaluation of these and many other factors—optical, climatic, thermal, economic, etc.—and it is unlikely that any single concentrator type will be desirable for all applications. 5.3.2 Thermodynamic limit of concentration The fundamental problem of radiation concentration can be stated as follows: How can radiation that is uniformly distributed over a range of angles \6\ < 1 6 a I and incident on an aperture of area Aap be concentrated on a smaller absorber area Aabs and what is the highest possible concentration
The problem as stated covers not only direct solar radiation, which is almost uniformly distributed over the solar disk with half-angle A5. = 4.7 mrad « 0.25°, but also the time-averaged radiation incident on a stationary aperture from a moving point source. In the latter case the radiation becomes partially diffuse when averaged over time. This consideration is important for the design of solar concentrators that do not require any tracking during a given period, for example, during the course of an entire day. Even though the problem of concentrating solar radiation appears to be one of geometric optics, it is instructive to examine first the limitations imposed by thermodynamics. There must be a connection between optics and the second law of thermodynamics, because if solar radiation could be concentrated onto an arbitrarily small receiver, the receiver temperature could exceed the surface temperature of the sun. This would obviously be a violation of the second law, which states that heat cannot flow from a cold surface to a hot surface without an external source of work. As a first step of the argument let us consider the exchange of radiation between two black surfaces of areas A{ and A2 of arbitrary geometry. For
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127
simplicity we assume that both are surrounded by vacuum; the generalization to radiating surfaces in a medium with refractive index n will be treated later. Suppose that A, and A2 have temperatures T1, and T2, respectively, whereas the rest of the universe is at absolute zero. The power radiated from each of these surfaces is
where a is the Stefan-Boltzmann constant
To find the net rate of radiative heat transfer between At and A2 we define ?,_ 2 as that fraction of the radiation emitted from surface At that reaches A2, either directly or via reflection(s) and/or refraction. If only direct radiation from AI can reach A2, then 57,_ 2 reduces to what is known in the heat transfer literature as radiation shape factor F, _ 2 and is given by the double integral
The shape factors have been calculated and tabulated for many configurations, and for simple geometries shorthand methods are available (e.g., see Sparrow and Cess [1978] or Siegel and Howell [1972]). See Fig. 5.3.1 for a diagramatic representation. In terms of the E? factors, the radiation emitted by A{ and absorbed by A2 is
Figure 5.3.1 Definition of radiation shape factor. Concentrator may contain reflective and/or refractive elements, and the surfaces may have any shape.
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Active Solar Collectors and Their Applications
with an analogous expression
for the radiation in the opposite direction. By the second law of thermodynamics, the net radiation transfer
must equal zero if the surfaces are at equal temperature. This implies the so-called reciprocity relation
for radiative exchange between any two surfaces. The reciprocity relation is independent of temperature, and it is a powerful tool for the analysis of radiation heat transfer (e.g., See Sparrow and Cess [1978] or Siegel and Howell[1972]). Now let us apply these concepts to solar concentrators. Any radiation concentrator can be represented by the configuration in Fig. 5.3.2. Since there are three surfaces involved, one can write down three reciprocity relations. The following two are of special interest:
in addition to the quantities denned above, As is the surface area of the radiation source (in our case the sun). These two equations can be combined to express the concentration C = A^/A^ in terms of J factors alone:
Figure 5.3.2 Schematic geometry of a radiation concentrator.
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The highest possible value of concentration corresponds to the case where all radiation from the source that enters the aperture goes on to the absorber. For this case energy conservation requires that
We further note from the geometry of Fig. 5.3.2 that 7 ap _, is just the fraction of radiation emitted by the aperture that reaches the source directly; hence it is a radiation shape factor:
With the help of Eqs. (5.3.14) and (5.3.15) the concentration of Eq. (5.3.13) can be stated in the form
Since the highest possible value of S7abs_s is unity,
we conclude that the thermodynamic limit of concentration is
For the special case of a source at infinity, placed symmetrically to subtend a half-angle da as seen from the aperture, the shape factor is
in two dimensions and
in three dimensions. The corresponding concentration limits are
and
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Active Solar Collectors and Their Applications
So far we have assumed the absorber to be surrounded by vacuum. Now suppose that the absorber is covered by a parallel slab of a transparent medium with index of refraction n. If the radiation incident on the slab is isotropic, then inside the slab it will be restricted to angles 10 in | < Bm where sin 9,, = \/n by Snell's law. Thus, further concentration by a factor l/(sin2 9,,) - n2 is allowed by Eq. (5.3.22), and the total concentration is bounded only by
There is no conflict between this value and the second law because an emitter in a medium of index n radiates n2 as much energy as an emitter in vacuum, a fact evidenced by the formula for the Stephan-Boltzmann constant (e.g., See Section 9.15 of Reif [1965]):
where k is Boltzmann's constant, h is Plank's constant, and c0 is the velocity of light in vacuum. An immediate corollary is that the increase in concentration brought about by a medium of index n > 1 does not reduce radiative losses (unless the medium has low thermal conductivity and is opaque to infrared). In general, the use of a medium with n > 1 for the purpose of increasing concentration is advisable only when high cost demands that the absorber area be as small as possible; this is relevant for photocells. In this book we treat only the case of n = 1, unless otherwise indicated. For imaging instruments, such as lenses, the light concentration properties are sometimes stated in terms of the .F-number, denned as ratio of focal length /and aperture diameter d:
Suppose the source subtends a half-angle 6 « 1. If the instrument is aberration free, it will form an image with diameter a = 2/0. This determines the size of the absorber area, giving a concentration
We see that the second law of thermodynamics implies a lower limit
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for the /"-number of an aberration-free optical instrument. Simple lenses and reflectors are not aberration free; e.g., parabolic reflectors do not hav a sharp focus for off-axis incidence. Hence the limit does not apply. In fact some reflectors used for solar energy have F-numbers below 0.5.
5.3.3 Acceptance of diffuse radiation Solar concentrators that are to require little or no tracking must have a fairly large acceptance angle, and therefore can collect a significant amount of diffuse radiation. A precise calculation of this effect requires detailed information about the angular distribution of diffuse sky radiation. Since at the present time hardly any data on this distribution are available, we shall simply assume that the hemispherical insolation is the sum of a beam component and an isotropic diffuse background. The fraction of the isotropic component 7d that is accepted by a concentrator is equal to S?ap-abs in the notation of the reciprocity relation
for the general concentrator geomentry shown in Fig. 5.3.2. In terms of th concentration C = Aap/Aabs this reads
In most solar concentrators, in particular V troughs and CPCs, all or almost all of the radiation emitted from the absorber reaches the aperture. Hence one can set yabs-ap — 1 and conclude that the fraction of isotropic radiation accepted by a solar collector of concentration C is
independent of the details of the concentrator. (For focusing parabolas with rim angle $ < 90° and without a second stage, and for Fresnel lenses without side reflectors, S7abs_ap can be somewhat smaller than 1, possibly as low as 0.5, and consequently the acceptance for diffuse radiation will be smaller. However, such concentrators would be used only in tracking systems with such high values of C that the contribution of diffuse radiation is negligible anyway.) Because of the predominance of near-forward scattering in the atmosphere, the sky radiation tends to be centered around the sun, and therefore the isotropic model underestimates the actual acceptance for diffuse radiation. Hence Eq. (3.2.4) for the solar radiation / accepted by a collector of
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geometric concentration C is actually a lower bound:
Data taken at Argonne National Laboratory by Reed [1976] and summarized in Table 5.3.1 indicate that 7 is not much larger than this lower bound. 5.3.4 Operating temperature as a function of concentration In this subsection we calculate the temperature at which a solar collector can operate as a function of concentration. Since we want to illustrate only certain gross features, we make some simplifying assumptions. To avoid the complication of multiple reflections we assume the sun and the rest of the universe to be blackbodies. The sun is at a surface temperature rsun; the rest of the universe (other than the sun and the absorber), collectively called ambient, is at ramb = 0. The radiation emitted by the sun is
where r is the radius of the sun. The radiation incident on a collector of aperture area Aap is
where R is the sun-earth distance. In terms of the half-angle As of the sun,
the radiation transfer from sun to receiver is
TABLE 5.3.1 Measured Data for the Acceptance of Diffuse Sky Radiation by CPCs of Various Concentration Ratios, and Comparison with Eq. (5.3.29) (adapted from Reed [1976]) Fraction of I/, accepted by CPC with concentration Cb Sky conditions Light haze/blue sky Heavy haze/white sky a
lk [W/m2]a
Ib/Ih
C = 10
C=5
C=3
1000
0.88
0.89 (0.892)
0.91 (0.904)
0.92 (0.920)
920
0.79
0.82 (0.811)
0.85 (0.832)
0.87 (0.860)
Pyranometer at normal incidence. The numbers in parentheses are based on Eq. (5.3.29) for isotropic sky radiation.
b
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where T = (1 — optical losses in atmosphere and in collector) and a is the absorptance of absorber for solar radiation. The radiation losses from the absorber are
where t is the emittance of the absorber in the infrared region corresponding to rabs. If a fraction 77 of the incoming solar radiation Qsm^bs is extracted a useful heat and/or lost by convection or conduction, then the energy balance of the absorber reads
or
Inserting the actual concentration C = Atp/Atbs and the ideal concentration Cidcai = I/sin 2 As, we find the operating temperature of the receiver to be
Since a and e become equal as Tabs approaches jTsun we see that the highest possible absorber temperature is !TabSi max = rsun « 6000 K, which can be reached only if no heat is lost or extracted and if the concentration equals
For two-dimensional concentrators,
and the absorber temperature is limited by Tsuri 213 selective absorbers (a = t).
1/4
« 1600 K for non-
5.4 IMAGE SPREAD DUE TO FINITE SOURCE AND OPTICAL ERRORS 5.4.1
Angular distribution of beam radiation
When viewed from the earth the sun appears as a disk of angular radius
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Active Solar Collectors and Their Applications
Seasonal variations in A v due to the excentricity of the earth's orbit are so small (± 1.7%) that they can be neglected. For most applications one need not worry about the detailed brightness distribution of the solar disk. Only collectors with very high concentration are sensitive to details of the angular brightness distribution of the sun. A typical solar brightness distribution is listed in Appendix E as B(8) [in W/m2-sterad] versus angular distance 6 from the center of the sun; it is also plotted in Fig. 5.4.1 in dimensionless form. This distribution is the simple average of all solar and circumsolar data obtained by the Lawrence Berkeley Laboratory circumsolar telescope [Rabl andBendt, 1982]. The distribution in Fig. 5.4.1 is seen to be fairly flat over the center of the sun, but it decreases near the edge, a phenomenon known as "limb darkening." At the edge the brightness drops sharply by several orders of magnitude. The region beyond 4.7 mrad and out to 50 mrad is called "circumsolar region." The limb darkening of the solar disk is caused by absorption and scattering in the photosphere of the sun and the atmosphere of the earth. If the sun radiated isotropically and if there were no narrow angle scattering in the atmosphere, then the brightness of the solar disk would be perfectly uniform. To understand why this is so, consider a flat blackbody surface element of area dA, tilted at an angle 9 from the line to the observer. This element radiates dq = oT4 dA cos 9 towards the observer. However, the observer sees only the projected area dAp - dA cos 0 of the element. The apparent brightness is the radiated power dq per apparent area dAp and equals crT4 independent of angle because dA cos 0 cancels out. Actual solar brightness distributions vary a great deal with atmospheric conditions; under hazy sky the radiation from the solar disk is less, the circumsolar radiation more than under a clear sky. Nevertheless the standard circumsolar scan of Fig. 5.4.1 (with separate weather dependent weighting
Figure 5.4.1 Brightness distribution (dimensionless) of standard solar and circumsolar scan. For a listing of the numerical values, see Appendix E (From Rabl and Bendt [1982]).
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factors for the solar and the circumsolar portions) is very useful because it permits a simple yet accurate calculation of the long term average effect of circumsolar radiation on the performance of highly concentrating collectors, as described in Section 8.6. For a statistical analysis of the radiation intercepted by a collector it is convenient to use the rms width of the sun rather than its radius. For pointfocus geometry (e.g., parabolic dish) the rms width
For line-focus geometry (e.g., parabolic troughs) one has to distinguish the angular variables 6l and 0j_ parallel and perpendicular to the direction of the absorber. They are indicated in Fig. 5.4.2. Only the direction perpendicular to the absorber needs to be considered for line-focus collectors because in the parallel direction all rays reach the absorber, no matter how large their angle 0,. Therefore the relevant rms width of the sun for line-focus geometry is given by
Since 02 = 0J + 0\, it is clear from Eq. (5.4.2) that
3 For simplicity the limit of integration has been extended to infinity because in practice 5(0) can be assumed to be negligible beyond a few degrees.
Figure 5.4.2 Field of view of (a) line-focus and pointfocus (b) collector aimed at solar disk. Shaded region indicates radiation that is not accepted by collectors. The orientation of the angular coordinates 0, and 0i parallel and perpendicular to the absorber in part (a) is also shown.
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hence the rms width of the sun for point-focus geometry is \/2 times the rms width for line-focus geometry
An intuitive explanation for this difference between line- and point-focus geometry is given in Fig. 5.4.2. The solid circle represents the source while the dotted lines show the field of view of a collector of high concentration with point focus (a) and line focus (b). As drawn in this figure the field of view characterized by an acceptance angle 26a is smaller than the width 2A, of the source. Thus the radiation from the shaded portion of the source is not collected. For the same value of 9a the portion that is lost is seen to be smaller for a line-focus collector than for a point-focus collector. In this sense the sun appears smaller to a line-focus collector than to a point-focus collector. If the sun were a uniform disk, the rms width would be
For real solar brightness distributions the rms width varies with atmospheric conditions (due to the contributions of circumsolar radiation). Under very clear sky (average clear sky) the width is
("Very clear sky" corresponds to circumsolar scan 1; "average clear sky," to the average over the rms widths for circumsolar scans 1 through 10 of Grether and Hunt [1977]; their /„ values range from 639-954 W/m2. A description of these scans can also be found in Biggs and Vittitoe [1979] and inRablandBendt[1982].) 5.4.2 Statistics of reflecting surfaces Snell's law of reflection is an idealization based on perfectly smooth surfaces. Real surfaces tend to have all kinds of irregularities, from microscopic roughness to macroscopic undulations. Furthermore, real reflector surfaces will not conform exactly to the design shape, and tracking collectors may have alignment errors. Finally the sun itself is not a point source but an extended source of radiation. All these effects contribute to enlarging the focal zone of a solar concentrator. For solar energy calculations a statistical analysis of these effects is adequate, because one is only interested in overall
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or averaged performance features, quite apart from the fact that it would be practically impossible to measure and monitor the reflector surface to its last detail. In discussing the features of reflector surfaces it is conceptually helpful to distinguish three scales as illustrated in Fig. 5.4.3. The dotted line shows the design shape, typically a parabola. The solid line averages over the undulations. The difference between the design shape (dotted line) and the average shape (dashed line) represents large scale optical errors, caused by gravity, wind, materials stress, or manufacturing errors. The waviness, with typical wavelengths on the order of centimeters to decimeters, represents medium scale errors. Finally, there is the small scale: microscopic surface roughness, which causes scattering away from the specular direction. This classification is not always clear-cut and in some cases there could be a continuous transition from small to large scale errors. Nonetheless, it seems to be useful for solar concentrators, and it corresponds to different measuring methods. Small scale errors are a property of the reflector surface itself whereas medium and large scale errors are due to the substrate and the support structure. Thus small scale errors are determined by measuring the spread of a collimated beam of light after it has been reflected from a small sample of the material. Medium and large scale errors can be determined by mapping the contour of the reflector using mechanical or laser ray tracing. Each of these optical errors can be characterized by a statistical distribution function. Only angular variations matter. Since image preservation is irrelevant, displacements in position need to be considered only to the extent that they cause deviations in angle. Ideally the distributions should be derived from measurements. From the available data it appears that the error distributions are usually well approximated by a Gaussian or normal distribution [Butler and Pettit, 1977]. Fortunately in many cases the details of the distributions do not matter even if they are not Gaussian. This follows from the central limit theorem of statistics, which says that the convolution of a large number of independent distributions approaches a Gaussian, even if the individual distributions are not Gaussian (See Cramer [1947] or Adams [1974]). Thus only the standard deviations of the individual distriFigure 5.4.3 Schematic sketch of large scale, medium scale, and micro scale errors of reflector surface. Dotted line shows design surface, dashed line shows average of actual surface, and solid line shows actual surface. Microroughness of surface causes scattering of reflected radiation over a range of angles.
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butions are needed, provided they have zero mean, as they usually do in solar applications. How to combine the individual error distributions will be the subject of Sections 5.4.4 and 8.3. For now we only look at the distributions associated with the microroughness of reflector materials. The lack of specularity of solar reflector materials have been investigated extensively by Pettit [1977]. Pettit found that for many reflector materials the scattering of radiation about the specular direction can be described by a Gaussian
where 0 is the angular deviation of a particular ray from the specular direction and R(6) is the intensity of radiation reflected into the direction 6. Rt is the total (i.e., hemispherical reflectance) and <TI is the width of the distribution. Some materials (e.g., Alzac, a rolled and polished aluminum sheet) exhibit different surface roughness in different directions. For these Pettit found it necessary to fit the data with a superposition of two Gaussians:
5.4.3 Effect of surface errors on width of reflected beam When designing solar concentrators one needs to know the effect of surface errors on the width of the reflected beam. Following Biggs and Vittitoe [1979] we consider a light ray that is incident in the direction i on a flat surface with surface normal n0. Let us choose coordinates such that the z axis is parallel to the surface normal n0,
and the incident ray lies in the (x, z) plane,
as shown in Fig. 5.4.4. Due to surface errors the actual surface normal n at the point of impact will differ by a small angle from the nominal direction A0.4 This angular deviation can be described by two angular coordinates: 6( measured in the x direction and e± measured in the y direction. e g is a rotation about the y axis; 4 Since the surface is continuous the position of the surface element will generally not be in the nominal reflector plane. However, since imaging and coherence properties are irrelevant for solar collectors, we can neglect these translational deviations and limit the analysis to angular deviations.
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Figure 5.4.4 Effect of surface contour error on angular deviation of reflected beam, i = direction of incident ray (in xz plane), A0 = nominal surface direction, ft = actual surface direction, ?0 = nominal direction of reflected ray, f = actual direction of reflected ray, e, and t± = angular deviations of surface, <5, and dL = angular deviations of reflected ray. ei is a rotation about the x axis. In the following we assume that the errors are small and we neglect all terms of order «2. Hence the actual surface normal is, to order e,
The distribution of surface errors can be characterized by a probability distribution E(et, «i) of zero mean whose width is given by
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Active Solar Collectors and Their Applications
One might naively expect the reflected beam to have widths 2al and 1<sL\ however, this is not quite true as we shall demonstrate by the following argument. First we calculate the direction r of the reflected ray according to Eq. (5.1.4):
The result is
where f a is the direction for no surface errors (i.e., for n = n0)
Then we express the difference between f 0 and f:
as an angular deviation from r0. For the angular deviation \ in the (x, z) plane we consider the projection of 5 onto the (x, z) plane:
The angular deviation St is given by
because 5 is small and perpendicular to f 0 . The result is readily found to be
In the direction perpendicular to 8XZ and to f 0 , the vector 8 has the projection
Hence the angular deviation perpendicular to 8XZ and f 0 is
or
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Equation (5.4.20) implies that the reflected beam deviation in the longitudinal direction equals the deviation of the surface times a factor of 2 for Snell's law, as expected:
However, in the perpendicular direction the reflected beam deviation is reduced by a factor cos 6, according to Eq. (5.4.23):
This result is not very intuitive, and it has in fact been overlooked in many discussions of focusing collectors. 5.4,44
Combination of optical errors
In a solar concentrator, several statistically independent factors contribute to the optical error: contour errors, lack of perfect specularity, tracking errors, and deformation and displacement of the receiver. Conceptually, concentrator alignment and tracking errors must be handled differently from mirror contour errors. Any given moment a concentrator is pointing in some direction, albeit with some angular error. It is only when an entire field of concentrators or a long time average is considered that a distribution in tracking error is obtained, described by a standard deviation
"contour is multiplied by 2 because of Snell's law; in Fresnel reflectors, o-tracking must also be multiplied by 2. Note that this rule for combining standard deviations is valid regardless of the shape of the individual error distributions; they could be Gaussian, boxlike, or anything else, since all distributions under discussion have zero mean. The total beam width o-tot is obtained by adding the rms width of the sun according to
Measurements of reflector surfaces by Butler and Pettit [1977] have shown that the distributions corresponding to
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Gaussian. Receiver displacements may be parallel or perpendicular to the aperture (or some combination), and the corresponding error distribution will in general not be Gaussian for a single collector module. Averaged over a large array of collectors, a Gaussian approximation for the displacement error distribution is likely, however, to be quite good. The other terms may or may not be Gaussian. However, when many statistically independent distributions are convoluted, the result is nearly Gaussian unless a single nonGaussian contribution dominates [Cramer, 1947; Adams, 1974]. In the case of focusing solar collectors, the Gaussian contour errors appear to be the largest, and a Gaussian approximation for the total optical error is usually reasonable. If a single Gaussian does not give an accurate enough representation of the sun, one can employ the formulation developed by Vittitoe and Biggs [1981] which treats the sun as a superposition of six Gaussian terms. 5.5 IMAGE SPREAD FOR LINE FOCUS COLLECTORS In Section 5.1 we showed how the optical analysis of troughlike concentrators can be simplified by projecting the three-dimensional ray trace diagram onto the plane perpendicular to the tracking axis. In the case of solar collectors a slight complication arises from the projection of the solar disk onto this plane. Since the sun moves about the earth in a circle, the projected width of the solar disk depends on time of day. This is illustrated in Fig. 5.5.1 for a parabolic trough with east-west tracking axis. The trough is located at the earth's surface at the origin of the x, y, z coordinate system in Fig. 5.5.1. The coordinates are fixed in the trough. The collector tracks about the z axis, to make sure that the sun is always in the y, z plane. Figure 5.5. la shows the projection onto the y, z plane; this plane contains the sun and the tracking axis. Figure 5.5. Ib shows the projection onto the x, y plane; i.e., the plane normal to the tracking axis. At noon the sun is in the xy plane and the angular half-width As of the solar disk (note A 3 « 1) is
with r the radius of the solar disk and R the distance from the earth to the sun. In the reference frame of the collector the apparent diurnal motion of the sun is a circle of radius R around the earth; therefore, away from solar noon, the projected angular half-width of the sun in the xy plane is
where Rxy is the projection
Fundamentals of Optics for Solar Collectors
143
Figure 5.5. 1 Position of sun relative to line-focus concentrator with east-west tracking axis at noon and in morning or afternoon, (a) Plane spanned by optical axis and tracking axis, (b) Projection on plane normal to tracking axis.
of the sun-to-earth distance on the xy plane. This is the projected angular width, which must be used as input in the two-dimensional ray trace diagram, and hence the rms width of the solar image on the receiver varies as
For a concentrator with east-west axis, the effective angular width of the sun at 4 hours from noon will be twice as large as at noon. For tracking polar-mounted concentrators, on the other hand, B equals the solar declination d, and this effect can probably be neglected because cos 5 is always larger than 0.92. For collectors with horizontal north-south tracking axis however, this image spread can cause serious problems at latitudes X far from the equator. Not only is the image enlarged by a factor I/cos (X + 5), but the summer-to-winter variation yields the widest image and hence the lowest intercept factor at a time when the insolation is also at its minimum. Combining the solar width with optical errors can now proceed as described in the last section. Suppose a line focus concentrator has a contour error distribution with rms width o-COntour ± but no other errors. (The subscript _i_ refers to deviations in the direction perpendicular to the tracking axis). Then the projected rms width of the reflected beam is 2
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ing standard deviations:
Only the width of the sun is enlarged by I/cos 9; the contour error remains the same. This" point has been missed by some publications on the subject. The fact that the path length between reflector and absorber increases like I/cos 0 has misled many people to the erroneous conclusion that the entire beam width broadens by the same factor. The I/cos 6 factor for the path length is canceled by the cos 0 that reduces the transverse width of the error term for the reflected beam in Eq. (5.4.25). In general there will be additional errors, due to imperfect specularity, tracking, and displacement of the receiver. Each of these is characterized by its rms width a, with appropriate subscripts. Specularity and contour errors can cause beam spreading both in the transverse and in the longitudinal direction. Longitudinal beam spread does not matter because in that direction the receiver is long and intercepts all rays anyway. Longitudinal surface errors do not contribute to any transverse beam spreading, at least as long as the reflector normal does not differ by a large angle from the optical axis. Hence longitudinal surface errors can be neglected for line-focus collectors with small rim angles (say < 45°), or equivalently large F-number (larger than about 1). In that case the beam spread from all optical errors has an rms width (r0plica] given by
This formula holds also for large rim angle (small ^-number) if the incidence angle is sufficiently small (say less than ± 45°); this is the case for polar mounted parabolic troughs. For parabolic troughs with large rim angles and incidence angles that are large during a significant part of the time (i.e., for east-west and for northsouth horizontal tracking axis) the longitudinal errors do make a noticeable although small contribution. A detailed analysis [Bendt et al., 1979] has recommended the rule
with
as a suitable year-round average. The numerical values of the factors in the last equation can be found in Table 5.5.1.
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Fundamentals of Optics for Solar Collectors
TABLE 5.5.1 Quantities Needed for Evaluation of Transverse Effects of Longitudinal Contour Errors in Parabolic Trough with East-West Tracking Axis (from Bendt et al. [1979]) Average over aperture Rim angle <j> (degrees)
0 30 45 60 75 90 105 120
2
(11,) aperture
Average over time of day 1
0 0.023 0.052 0.093 0.147 0.215 0.297 0.395
2
2tan(0/2)
Cutoff time 4,00
0 3 4 5 — — —
—-
„,,
0 0.2 0.5 0.9 — — — —
The total beam spread
REFERENCES Adams, W. J. 1974. The Life and Times of the Central Limit Theorem. New York: Kaedman. Bendt, P., Rabl, A., Gaul, H. W., and Reed, K. A. 1979. "Optical Analysis and Optimization of Line Focus Solar Collectors." Report SERI/TR-34-092. Golden, CO: Solar Energy Research Institute. Biggs, F. and Vittitoe, C. N. 1979. "The HELIOS Model for the Optical Behavior of Reflecting Solar Concentrators." Report SAND 76-0347. Albuquerque, NM: Sandia National Laboratories. Born, M. and Wolf, E. 1975. Principles of Optics, 5th ed. New York: Pergamon. Butler, B. L. and Pettit, R. B. 1977. "Optical Evaluation Techniques for Reflecting Solar Concentrators." Optics Applied to Solar Energy Conversion. Society of Photo-Optical Instrumentation Engineers, Vol. 114, p. 43. Cramer, H. 1947. Mathematical Methods of Statistics. Princeton, NJ: Princeton University Press. Dufne, J. A. and Beckman, W. A. 1980. Solar Engineering of Thermal Processes. New York: John Wiley & Sons. Edwards, D. K. 1977. "Solar Absorption by Each Element in an Absorber-Coverglass Array." Solar Energy, 19:401. Grether, D. F. and Hunt A. 1977. "Description of the LBL Reduced Data Base." Informal Report. Berkeley, CA: Lawrence Berkeley Laboratory. Pettit, R. B. 1977. "Characterization of the Reflected Beam Profile of Solar Mirror Materials." Solar Energy, 19:733. Rabl, A. 1976. "Comparison of Solar Concentrators." Solar Energy 18:93. Rabl, A. 1977a. "Radiation Transfer Through Specular Passages—A Simple Approximation." Int. J. Heat and Mass Transfer 20:323. Rabl, A. 1977b. "A Note on the Optics of Glass Tubes." Solar Energy 19:215.
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Rabl, A. 1980. "Concentrating Collectors." In Solar Energy Technology Handbook, Dickinson, W. C. and Cheremisinoff, P. N. editors. New York: Marcel Dekker. Rabl, A. and Bendt, P. 1982. "Effect of Circumsolar Radiation on Performance of Focusing Collectors." ASME J. Solar Energy Eng., 104:237. Reed, K. R. 1976. "Instrumentation for Measuring Direct and Diffuse Insolation in Testing Thermal Collectors." Optics in Solar Energy Utilization II. Society of Photo-Optical Instrumentation Engineers, Vol. 85. Reif, F. 1965. Fundamentals of Statistical and Thermal Physics. New York: McGraw-Hill. Siegel, R. and Howell, J. R. 1972. Thermal Radiation Heat Transfer. New York: McGraw-Hill. Sparrow, E. M. and Cess, R. D. 1978. Radiation Heat Transfer. New York: Hemisphere Publ. Corp. McGraw-Hill. Vittitoe, C. N. and Biggs, F. 1981. "Six-Gaussian Representation of the Angular Brightness Distribution for Solar Radiation." Solar Energy 27:469. Welford, W. T. and Winston, R. 1978. The Optics of Nonimaging Concentrators. New York: Academic Press. Winston, R. 1970. "Light Collection Within the Framework of Geometrical Optics." /. Opt. Soc. Am., 60:245.
6. OPTICS OF NONTRACKING COLLECTORS
The principal nontracking collector types are flat plates, flat plates enhanced by side reflectors or V-troughs, tubular collectors, and compound parabolic concentrators (CPC). Apart from trivial shading and end effects flat plates require no optical analysis other than the calculation of the transmittanceabsorptance product, which has already been discussed in the previous chapter. As for the V-troughs and side reflector enhancements, they can be considered as straight-line approximations to CPCs. For a given relation between aperture and radiation source the highest possible concentration is achieved by a CPC. Hence it is instructive to begin this chapter with a fairly detailed description of the large class of CPC configurations. The next section discusses reflector configurations for evacuated tubes. After that, we address the important practical case of V-troughs and side reflectors. 6.1 COMPOUND PARABOLIC CONCENTRATORS
Concentrators that reach the thermodynamic limit of concentration
for an acceptance half-angle 6a, have been called ideal concentrators because of their optical properties. In the solar energy literature, names such as compound parabolic concentrators (abbreviated CPC) and nonimaging concentrators have also been used. We shall refer to all concentrators of this class as CPCs, even though some of them are not even parabolic in shape. Ideal concentrators are a surprisingly recent discovery. The first example of a CPC, shown in Fig. 6.1.1, was found independently in the United States by Hinterberger and Winston [1966], in Germany by Ploke [1966], and in the U.S.S.R. by Baranov and Melnikov [1966]. It consists of parabolic reflectors that funnel the radiation from aperture to absorber. The right and left halves belong to different parabolas, as expressed by the name CPC. The axis of the right (left) parabola makes an angle + 6a (— 6a) with the collector 147
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Figure 6.1.1 Compound parabolic concentrator (CPC).
midplane, and its focus is at A (B). At the end points C and D, the slope is parallel to the collector midplane. Tracing a few sample rays reveals that this device has the following angular acceptance characteristic: all rays incident on the aperture within the acceptance angle, that is, with 19 \ < 6a, will reach the absorber, whereas all the rays with 16 \ > 6a will bounce back and forth between the reflector sides and eventually reemerge through the aperture. This property, plotted schematically by the solid line in Fig. 6.1.2, implies that the concentration is equal to the thermodynamic limit.
Figure 6.1.2 Fraction of the radiation incident on aperture at angle 6 that reaches absorber, for CPC with acceptance half-angle 6a, assuming reflectivity p = 1. Solid line: untruncated CPC with perfect reflectors; dashed line: truncated CPC with perfect reflectors; dotted line: untruncated CPC with surface errors a (From Rabl [1976a]).
Optics of Nontracking Collectors
149
Subsequent to the discovery of the basic CPC, Fig. 6.1.1, several generalizations of the ideal concentrator have been described that are relevant to special applications. These generalizations concern the following: 1. The use of arbitrary receiver shapes [Winston and Hinterberger, 1975; Rabl, 1976]; e.g., fins and tubes, see Fig. 6.1.3. (The latter are important because of their ability to carry a heat transfer fluid.) 2. The restriction of exit angles 0OU, at the receiver [Rabl and Winston, 1976] to values 10OU, | < B2 < ir/2; see Fig. 6.1.4. It is important because some receivers have poor absorptance at large angles of incidence. It is also needed for the design of CPCs that function entirely by total internal reflection [Winston, 1976]. 3. The asymmetric orientation of source and aperture; see Fig. 6.1.5 (for the design of collectors with seasonally varying outputs [Rabl, 1976a; Mills, 1978].) 4. The matching of a CPC to a finite source of radiation; see Fig. 6.1.6 (useful as second-stage concentrators to collect radiation from a first stage which is a finite distance away). The design goal is to concentrate radiation maximally, subject to any of
Figure 6.1.3 CPCs with different absorber shapes. All have the same absorber perimeter a and acceptance half-angle Ba.
Figure 6.1.4 6, — 02 transformer, consisting of parabolic sections P« and PL and of straight sections SR and SL. The slope 7 of the straight section is 7 = (02 — 0i)/2. 0, = 8a is the acceptance halfangle (From Rabl and Winston [1976]).
Figure 6.1.5 Asymmetric CPC with acceptance angle 26a =
150
Optics of Nontracking Collectors
151
Figure 6.1.6 Exemplary design of an ideal concentrator for finite source. Radiant energy from diffuse source (S) enters aperture BB' and is maximally concentrated onto the absorber (K). The angle of incidence on R is restricted to 10oul | < 62. A typical extreme ray originates from A', impinges on the aperture with angle of incidence 0in, and, after one reflection, is directed to point P on the absorber with angle of incidence 02 (From Rabl and Winston [1976]).
the subsidiary conditions 1 through 4 that may have been specified. The design of two-dimensional concentrators is determined uniquely by the extreme rays, extreme rays being denned as rays coming from the edge of the source. In three dimensions (geometry of cones and pyramids), the design is in general overdetermined, but at least for flat receivers a good compromise is achieved by choosing a surface of revolution whose cross section has been determined by the two-dimensional solution [Winston and Welford, 1978]. To describe the extreme ray design principle more fully let us consider concentrators that use only reflective elements and let us demand that the average number of reflections be as small as possible. Then the solution is to maximize the slope of the profile curve of the mirror, subject to the condition that extreme rays illuminate the absorber within the prescribed angular limits ± 02. This implies that rays originating at angle 02 from a convex absorber (flat absorbers are treated as limiting cases of convex absorbers, and in concave absorbers the cavities are replaced by flat chords stretched across cavity openings) emerge from the concentrator as extreme rays after undergoing at most one reflection.1 The procedure is illustrated in Fig. 6.1.6. Radiation emanating from the diffuse source S and entering the aperture BB' is concentrated onto the absorber R. The mirror curve must have such a slope that a ray emitted from any point P of the absorber at an angle + 62 'With tubular absorbers an exact solution is known only for the case D2 = ir/2; i.e., isotropic illumination at the absorber. However, one obtains a solution that restricts most rays at the absorber within 62 < *"/2 if one simply enlarges the radius of the absorber by a factor I/sin 02 beyond the 62 = ir/2 case, keeping the center of the absorber fixed. This solution works well in practice and is recommended whenever one wants to avoid very large incidence angles at the absorber (and the associated poor absorptivity of most coatings). Oversizing the absorber in this manner also increases the tolerance to mirror errors and errors in absorber placement.
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Active Solar Collectors and Their Applications
(~ 02) may be reflected toward the edge A'(A) of the source. The mirror starts at the edge C(C') of the absorber, and it terminates at the intersection B(B') with the limiting rays AC'(A'C) to avoid shading the absorber. Then the geometric concentration is given by [see Eq. (5.3.23)]
where n is the index of refraction of the medium surrounding the absorber and F ap _ s is the radiation shape factor for diffuse radiation from aperture to source. Speaking mathematically, the concentrator slope is uniquely determined by a first-order differential equation with one boundary condition (the reflector curve must pass through the edge of the absorber). For the important example of a tubular absorber we present the explicit equation of the CPC reflector. It is convenient to use the special coordinates of Fig. 6.1.7. The point B of the reflector is described in terms of two parameters: the angle 6
subtended by the lines from the origin O (center of the absorber tube) to A and C, and the distance
Figure 6.1.7 Coordinates and parameters for description of CPC for tubular absorber.
Optics of Nontracking Collectors
153
BC being tangent to the absorber tube at C. The solution is given in two separate portions, an ordinary involute for 10 \ < 6a + ir/2, and an outer portion for 16 \ > 6a + ir/2:
r is the radius of the absorber tube. As for the choice between different CPC types, the configurations with fin or tube absorbers, Figs. 6.1.3b and 6.1.3d are preferable for most solar applications. Not only is the absorber material used more efficiently than in other designs, but heat losses through the back are low. This may be quite an important advantage, because it may not be cost effective to reduce the effective U value of the back of a collector much below 0.5 W/m2 K. Compared to a frontal U value on the order of 3 W/m2 K [Rabl et al., 1980] for threefold and 1.4 W/m 2 K for tenfold concentration, the losses through the back are indeed significant. Thus the reduction in back losses possible with the configurations of Figs. 6.1.3b and 6.1.3d will more than compensate for the lightly higher optical losses (the average number of reflections for the configurations of Figs. 6.1.3.b and 6.1.3d is approximately 0.5 higher than for the configuration of Fig. 6.1.3a, as shown in Figs. 6.1.10a and 6.1. lOb). An asymmetric CPC is shown in Fig. 6.1.5 [Rabl, 1976a]. Its acceptance angle is 26a = / + „ and its geometric concentration is
At central incidence the effective concentration, including cosine factor, is I/sin da. Asymmetric CPCs may be practical if the reflector can be made at very low cost compared to the cost of the absorber [Mills 1978]. In their optical properties all CPC types are exactly or almost exactly alike. Above all, the same relation, Eq. (6.1.1), exists between their concentration and angular acceptance, with the sharp cutoff implied by Fig. 6.1.2. The flux distribution at the absorber depends on the angle of incidence and on absorber shape, and must be determined by detailed ray tracing. However, the following important statement can be made about all CPCs, without any need for ray tracing: If the radiation incident on the aperture is spread uniformly over the entire acceptance angle, then it will be isotropic when it reaches the absorber—unless the design was chosen to restrict the exit angles to values below 82 < w/2, in which case the radiation at the absorber will uniformly fill the angular range from — 82 to +62. This consideration of uniform illumination is very important because it gives a simple and reliable estimate of the average performance of a CPC solar collector. When beam insolation is incident at certain angles, hot spots of high flux concentration (up to about 40) may appear on the absorber.
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Active Solar Collectors and Their Applications
CPCs have large reflector areas. Fortunately, this disadvantage can be alleviated by truncation: the top portion of a CPC can be cut off with little loss in concentration since it does not intercept much radiation. Figure 6.1.8 shows the effect of truncation on concentration and on the ratio of reflector over aperture area for several design acceptance half-angles 8a. Similarly, Fig. 6.1.9 shows the effect of truncation on the depth (expressed as heightover-aperture-width ratio) of a CPC. Both figures were calculated for the CPC with flat one-sided receiver, but they are approximately correct for other CPC types. Analogous graphs for CPCs with tubular absorbers have been published by Mclntire [1979]. The number of reflections varies both with angle of incidence 6 and with point of incidence on the aperture. A very good estimate of the fraction of radiation transmitted through a CPC with wall reflectivity p is given by the simple formula
where {«) is the average number of reflections; the validity of this approximation has been demonstrated by Rabl [1977]. For incidence at one particular angle 0 one needs {«(#)}, the average over all incidence points at angle d, whereas {«), the average of («(0)) over all 6 within the acceptance angle, is relevant for the average performance. Figure 6.1.10 shows («), along with the high and low values of {«(#)) for several acceptance angles (reflector profiles), as a function of concentration and truncation. The variation of (n(8)) with 0 decreases with truncation. This feature is important because small variation is desirable for the sake of uniform collector output.
Figure 6.1.8 Reflector/ aperature ratio as function of concentration for full and truncated CPCs of configuration in Fig. 6.1.3a. Oa = acceptance half-angle (From Rabl [1976b]).
Figure 6.1.9 Height/ aperture ratio for full and truncated CPCs of configuration in Fig. 6.1.3a, as function of concentration and acceptance half-angle 6a (From Rabl [1976b]).
Figure 6.1.10 Number of reflections for full and for truncated CPCs computed by ray tracing. The average over all points of impact was taken at each angle of incidence 8 in order to find {«(#)}. For each of the acceptance half-angles 6a in this graph; the high and low values of (n(6)) are shown in addition to the avearge <«} over all |0| < Oa. (a) CPC with flat one-sided receiver (Fig. 6.1.3a). (b) CPC with fin receiver (Fig. 6.1.3b). Average number of reflections is approximately the same for fin configuration (Fig. 6.1.3b) and for tube configuration (Fig. 6.1.3d). For example, if a CPC with 8a = 11.5° and flat one-sided receiver is truncated to a concentration of 4, the average number of reflections ranges from a low of 0.76 to a high of 0.86 with a mean of 0.82.
155
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Active Solar Collectors and Their Applications
In practice one can often neglect the variation of («(0)} and simply work with the average (n) over all incidence angles. For a large class of concentrator and radiation passages (n) can be calculated in closed form, as shown by Rabl [1977]. Figure 6.1.10a is based on the standard CPC with flat onesided receiver. If fin or tube absorbers are employed, (n) will be higher by about 0.5 for concentration and truncation values of practical interest; i.e., C between 1.5-10. This is shown in Fig. 6.1.1 Ob. If a CPC is truncated, some rays outside the acceptance angle (10 \ > Oa) can reach the absorber, whereas of course no rays with 10 \ < da are rejected. The resulting increase in angular acceptance is, however, insignificant in most practical applications, as shown in Fig. 6.1.2. For example, if direct sunlight enters a CPC of Fig. 6.1.1, moderately truncated to a concentration C', with 0 > 8a, the fraction of radiation reaching the absorber is less than 1/C'; under these conditions the collector is useless for thermal and marginal for photovoltaic applications. The fraction of isotropic radiation that is accepted is of course 1/C' independent of any details of the concentrator. Regarding sensitivity to mirror surface errors, the analysis is equally simple for all CPCs, because their geometry implies that all rays incident near the cutoff angle, that is, with 18 \ < da, undergo exactly one reflection on their way to the absorber. In almost all practical applications, the acceptance half-angle Oa will be larger than 5°, and the mirror surface errors a will usually be small compared to da. Therefore, all of the rays with 101 < 0a — 2a and none of the rays with 101 > 6a + 2a will reach the absorber, whereas in the transition region, 8a — 2a < 101 < Ba + la, some rays are accepted and some are rejected. The resulting angular acceptance is shown schematically by the dotted line in Fig. 6.1.2. Further details on optical and geometric properties of CPCs can be found in Winston and Welford [1978]. The involute with unit concentration (C = 1) is included in this chapter as a special case of the CPC. The acceptance half-angle is 90°. The examples in Fig. 6.1.11 distribute incident radiation over the surface of a tubular or fin absorber, and the average number of reflections is 7r/4 for isotropic radiation [Rabl, 1977]. Recently some new concentrator types have been discovered that reach or closely approach the thermodynamic limit, even though they are quite different from the CPC. Winston and Welford [1979] discovered a generalized design principle that yields not only the CPC but also a novel type of second stage concentrator. The latter, called "trumpet" because of its shape, is described and evaluated in Section 7.6. An interesting prismatic reflector with total internal reflection has been described by Mills and Giutronich [1978]. It closely approaches the thermodynamic limit and is well suited as second stage concentrator for line-focus photovoltaic collectors [Mills, 1981]. To close this section we point out that the optical designs described here have many applications beyond solar energy. They are useful wherever one wants to redistribute radiation in a particular manner. The reflector of a photographic flash tube, for instance, is to distribute the light from the tube
Optics of Nontracking Collectors
157
Figure 6.1.11 Involute with unit concentration ratio, for tubular absorber (a), and for flat two-sided absorber (b).
in such a way as to uniformly illuminate the field of view of the camera. The appropriate design is a CPC or CPC-like lens-mirror combination. (The flash light industry seems to have discovered these principles independently). Obvious energy conserving applications lie ahead in the lighting industry when one wants to maximize the transmission of light from fluorescent tubes at the ceiling to the room below. 6.2 REFLECTORS FOR EVACUATED TUBES By and large the cost per unit area of reflectors is lower than the cost of evacuated tubes. For that reason most of the evacuated collectors sold today use some kind of reflector enhancement. Because of imperfect reflectivity the use of a reflector incurs some optical losses. At the same time heat losses are reduced because there is less absorber area per aperture area. Thus the choice of a reflector involves tradeoffs between optical performance, thermal performance, and cost. In general one wants a large acceptance angle to maximize collection time and/or minimize the need for tilt adjustments. From an optical point of view the CPC is the best reflector, but in some cases one might for practical reasons choose a different reflector such as a V-groove. The choice of the reflector depends on the shape of the absorber. Figure 6.2.1 shows several reflector arrangements that have been used with tubular
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Active Solar Collectors and Their Applications
Figure 6.2.1 Several reflector configurations for evacuated tubes, (a) Diffuse reflector, (b) V-groove reflector, (c) circular cylindrical reflector, (d) involute reflector (CPC) (From O'Gallagher et al. [1980]).
absorbers. The diffuse reflector in part a is just a plain white surface behind the tubes. It has the lowest cost and the lowest performance since much of the reflected radiation misses the tubes. V-groove and circular cylindrical reflectors in Fig. 6.2.1b and 6.2.Ic are also quite easy to fabricate, and in small quantities their cost can be significantly lower than the cost of CPCs. V-groove or V-trough reflectors are fairly well matched to flat absorbers, but with tubular absorbers they do not utilize the back of the tubes very well. The best optical performance is achieved with a CPC reflector, as in Fig. 6.2. Id. With mass production technologies the shape of the reflector has little influence on the cost, and it becomes advantageous to use the best shape. One way to describe the superior optical performance of the CPC is the following. With a CPC the entire aperture area is effective in directing radia-
Optics of Nontracking Collectors
159
tion to the absorber, over the entire range of incidence angles within the acceptance angle. This is illustrated by the photograph in Fig. 6.2.2. With the absorber tube in the design position of a CPC reflector trough the entire aperture appears black if viewed from any angle within the acceptance angle. An analogous photograph taken of the V-groove or circular reflector arrangements would reveal large shining portions in the aperture where incident radiation misses the absorber. At particular incidence angles the entire aperture may be active even with these reflectors, but only with a CPC is the entire aperture active over the entire range of incidence angles that has been specified. It is worth pointing out that in some cases one may want to design for a geometric concentration ratio below unity. Some absorber coatings of evacuated tubes have a rather low absorptivity. Together with reflections off the glass envelope these tubes may reflect 20%-30% of the incident radiation. If the reflected rays have another chance to hit a tube the overall effective absorptivity of the collector is improved. This kind of absorption enhancement is possible if the geometric concentration ratio is less than unity. In effect the absorber tubes plus their reflectors can act like a radiation cavity whose absorptivity exceeds the absorptivity of the absorber surface itself. This question has been studied by O'Gallaher et al. [1980], and the involute (CPC with concentration less than unity) was shown to give the greatest possible absorption enhancement. An important design problem for thermal CPC collectors arises from the need for a gap between reflector and absorber.2 The design principles for CPC reflectors demand that the reflector extend all the way to the absorber. However, this is not possible with evacuated tubes because of the finite thickness of the glass and because a finite spacing between absorber surface and glass envelope. Several modifications of the basic CPC design are possible to deal with this gap problem. One could, for example, leave the reflector intact and reduce the absorber size, or one could leave the absorber as it is and truncate the adjacent reflector. These and several other solutions have been investigated and the following can be recommended as the best, in terms of minimi/ing the optical loss. For flat absorbers the optical loss of the gap is minimized if the reflector is truncated adjacent to the absorber, as sketched in Fig. 6.2.3. As shown in Rabl et al. [1979], the optical loss for uniform illumination within the acceptance angle can be calculated in closed form using radiation shape factors. This is an excellent approximation for the yearly average loss in actual operation. For small gaps in the configuration of Fig. 6.2.3 the fraction of the incident radiation that is lost in the gap is given by
2 A gap is also needed in nonevacuated CPC collectors if the reflector has a high conductivity; otherwise the reflector may turn out to act as cooling fin for the absorber.
Figure 6.2.2 Photograph of CPC troughs with 35° acceptance half-angle. Evacuated absorber tube held in design position makes the entire aperture look black if viewed from anywhere within the acceptance angle (Courtesy of Argonne National Laboratory).
160
Optics of Nontracking Collectors
161
Figure 6.2.3 Truncation of CPC to accommodate gap between flat one-sided absorber and reflector (From Rabl et al. [1979]).
where
Depending on the acceptance angle 0a, the factor 1 + cos y ranges from 0.3-1.0. The gap loss with CPC is actually much smaller than it would be in a corresponding absorber array without CPC. If evacuated tubes with flat absorber of width a are placed next to each other, with gap of width g at each side, then the gap loss is approximately 2 g/a, about 4 times as large as with a typical CPC. For tubular absorbers one could also truncate the reflector, as suggested in Fig. 6.2.Id. The associated gap loss has been calculated by Rabl et al. [1979] and turns out to be quite small. However, with tubular absorbers one can do even better by avoiding gap losses altogether. This possibility was suggested by Mclntire [1980b] and explored systematically by Winston [1980]. This design replaces the cusp by a grooved cavity as shown in Fig. 6.2.4. The number and the dimensions of the grooves depend on the gap width. With a single groove the gap can be as large as 0.27 times the tube radius. A design with two grooves, shown in Fig. 6.2.5, permits a lossless
Figure 6.2.4 Modification of CPC for tubular absorber with finite gap g and zero optical loss. For the example chosen, the gap is g = 0.8 r, the groove angle 2^ = 118°, the acceptance half-angle 6a = 60°, and the concentration ratio C - 1.0 (From Winston [1980]).
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Active Solar Collectors and Their Applications
Figure 6.2.5 Gap-lossless reflector design allowing the largest gap for a two-groove reflector (R2 = 1.49^,). The dotted curve is a five-groove design, allowing a gap up to R2 — 1.74 /?, (From Mclntire [1980b]).
solution up to g = 0.4r. With a larger number of grooves the lossless design can be extended up to a maximum gap equal to the radius. The lossless design necessarily entails a certain sacrifice of concentration ratio. For instance the design of Fig. 6.2.4 achieves only a concentration of 1.0 even though the thermodynamic limit for this acceptance angle is I/sin 60° = 1.15. In most solar applications high optical efficiency is more important than the attainment of the highest possible concentration ratio; hence one will usually choose the lossless solution. The design principle of this solution has been described by Winston [1980] and is explained here by means of Fig. 6.2.4. The grooves are located on a circle around the center of the absorber, leaving a gap of width g. The portions AF and BE of the reflector form a CPC matched to an absorber ADCB (i.e., an absorber consisting of the top portion DC of the real absorber and of cavity openings AD and BC). For a valid design the cavity opening AD plus BC must be smaller than the bottom portion CD of the absorber. To ensure that all rays entering the cavity openings end up at the absorber one must choose the right dimensions for the V-grooves. Also the V-grooves must extend far enough to avoid any direct radiation from AD to BC. Since there is no need to extend the grooves any further than this, one will choose the points A and B to lie on or close to the tangent to the absorber. To find the proper dimensions of the V-grooves one starts out with a specified gap width g. With groove depth h and absorber radius r the angle 2« subtended by the absorber as seen from the edge of the groove is given by
Optics of Nontracking Collectors
163
as shown in Fig. 6.2.6. The angle a must not exceed 30°. The opening angle 2^ can be chosen somewhat arbitrarily provided it satisfies the constraints
Finally the groove height h can be chosen arbitrarily as long as it is below the upper bound.
For ease of manufacture one will probably want to keep the number of grooves as small as and the height h as large as possible. The limits on groove number as function of gap width have already been stated above. For a single groove the design procedure is slightly different because the quantity g in Fig. 6.2.6 will then be chosen as g = 0. The relevant gap parameter is now 8, the minimum distance between tube and the sides of the V-groove
It must satisfy the bound
Figure 6.2.6 Dimensions of the individual reflector grooves for the CPC modification without gap loss.
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Active Solar Collectors und Their Applications
The constraints in Eqs. (6.2.3) and (6.2.4) must also be satisfied. The largest gap 5 allowed with a single V-groove corresponds to 5 « 0.27r, a = 38.2° and i/' ss 51.8° [Winston, 1980]. Even larger gaps can be accommodated with a single V-groove while incurring only negligible losses [Collares-Pereira and Mendes, 1982]. The V-groove cavity design can also be useful in other applications. Figure 6.2.7 shows a V-groove cavity second stage concentrator for a parabolic trough with tubular absorber. This second stage design is appropriate for fairly large rim angles, around « 70°. For a detailed optical analysis of evacuated tubes with reflectors the reader is referred to articles by Mclntire [1980a] and by Window and Basset [1981]. 6.3 V-TROUGHS V-troughs are a classic concentrator design for applications with large acceptance angle; e.g., see Tabor [1958] and Hollands [1971]. To analyze the multiple reflections in V-troughs the method of images is convenient. The angular acceptance of the V-trough of Fig. 6.3. la is shown schematically by the solid line in Fig. 6.3. Ib. Compared to a CPC of the same concentration C = I/sin (^ + 0i) (see dotted line of Fig. 6.3. Ib), the useful acceptance angle #! of a V-trough is significantly smaller. Only when the trough angle ^ approaches zero does the acceptance angle approach that of the CPC; however, in this limit the trough becomes too deep and the reflection losses become excessive. The higher the desired concentration, the greater the relative advantage of the CPC over the V-trough. The upper limit of concentration for a practical V-trough is about 3 (as a nontracking collector with daily tilt adjustments). With summer/winter adjustments only, the V-trough is limited to concentration values below 2, and for a completely fixed collector a Vtrough gives almost no concentration. As for absorber shapes, the V-trough is limited to flat one-sided absorbers. An interesting asymmetric V-trough
Figure 6.2.7 V-groove cavity as second-stage concentrator for parabolic trough with large rim angle (From Mclntire [1980b]).
Optics ofNontracking Collectors
165
Figure 6.3.1 (a) V-trough concentrator with aperture A and Aabs. Mirror images and reference circle are also shown. The rays P, and P2 have angle of incidence 0, and 02, respectively; they pass through the edge of the absorber and are tangential to the reference circle, (b) Angular acceptance of V trough (schematic, neglecting difference between polygon and circle in [a]) (From Rabl [1976a]).
that requires summer-to-winter adjustment has been described by Selcuk [1979]. 6.4
SIDE REFLECTORS
Side reflectors can be a useful means of increasing the output of a flat plate collector. Especially in situations where a suitable surface is available right next to a collector it can be quite cost effective to add a reflector, for instance a sheet of anodized aluminum. For instance, if a tilted collector is mounted on a flat horizontal roof one might place a reflector in front. In large installations several rows of collectors may be mounted one behind another; in this case one may put reflectors behind the collector rows creating a saw tooth pattern of alternating collector and reflector surfaces. If a separate support structure is needed for a side reflector, the cost may be too high to be practical. Flat plates with a side reflector tend to produce a rather nonuniform output over the course of the year, just like an extreme asymmetric (or onesided) CPC [Rabl, 1976a] to which it is so-to-speak a straight-line approximation. Whether such nonuniform output is desirable depends on the load
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Active Solar Collectors and Their Applications
Figure 6.4.1 Flat plate collector (KLMN) with side reflector (ABCD) (From Grassie and Sheridan [1977]).
distribution. Depending on the geometry, the reflector may at some times of the year cast a shadow on the collector and actually reduce the performance. There is one case where a reflector is guaranteed to improve the collector output. A single tilted collector always receives some radiation from the ground during daylight hours; hence a horizontal reflector in front of the collector can only help. Whether it is cost effective is of course another question. As for the choice between a specular and a diffuse reflector, a diffuse side reflector has very little effect as shown by Grassie and Sheridan [1977]. The side reflector should be reasonably specular and its reflectivity should be as high as possible. To evaluate the benefit of side reflectors one must look at system performance, not just the optical enhancement. Since the heat loss is the same with and without reflector the increase of the collector output can be significantly higher than the optical enhancement factor. Apart from these simple facts it is difficult to make general statements about side reflectors. A detailed evaluation is needed in each case because the benefit of side reflectors depends on a large number of variables, not only the geometric/optical parameters but also the system configuration and load. Even the optical analysis of the reflector/collector combination is fairly complicated. The number of variables involved is indicated in Fig. 6.4.1. Both reflector and collector have finite length, and for the incident direction
Figure 6.4.2 Image and effective aperture for reflector/ collector combination.
Optics of Nontracking Collectors
167
Figure 6.4.3 Percentage increase of energy delivered by a hot water system due to the addition of a reflector above the collector, at tilt angle S, from horizontal (From Grassie and Sheridan [1977]).
of beam insolation in this figure the portion (KLEA'F) of the collector receives reflected radiation. For certain incidence angles the reflector may cast a shade: if the shadow of the corner A falls on A', then the entire portion KLEA'F is shaded. The reflector in Fig. 6.4.1 is above the collector; one could also place it below. The analysis is much simpler if the reflector is long compared to the collector because then the problem reduces to two dimensions. It is instructive to look at the two-dimensional case and add the image of the collector in the reflector, as shown in Fig. 6.4.2. For each incidence angle the effective aperture is the projected aperture of the collector plus the projected aperture of the image. How the addition of a side reflector might affect the system performance is illustrated in Fig. 6.4.3. In this particular example a domestic hot water system uses a flat plate collector at tilt 35° due north in Brisbane, Australia at latitude 27.5° south. Figure 6.4.3 displays the monthly performance change of the system due to the addition of a side reflector above the collector. Three different curves are shown, corresponding to three different reflector angles Sr. For Sr = 90° the delivered energy is always increased. For Sr = 70° the output is highly peaked in summer while it drops to 40% below the no-reflector case in winter. For more specific results (in particular, for the optical performance), the reader is referred to detailed studies in the literature [McDaniels et al., 1975; Seitel, 1975; Larson, 1980; Chiam, 1981]. REFERENCES Baranov, V. K. and Melnikov, G. K. 1966. Sov. J. Opt. Techn. 33:408. Chiam, H. F. 1981. "Planar Concentrators for Flat Plate Solar Collectors." Solar Energy 26:503. Collares-Pereira, M. and Mendes, J. F. 1982. "1.5X CPC Concentrator for High Temperature Applications with Cavity Enhancement Effect" SSPS Mid-Temperature Workshop, Almeria, Spain.
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Grassie, S. L. and Sheridan, N. R. 1977. "The Use of Planar Reflectors for Increasing the Energy Yield of Flat Plate Collectors." Solar Energy 19:663. Hinterberger, H. and Winston, R. 1966. Rev. Sci. Instrum. 37:1094. Hollands, K. G. T. 1971, "A Concentrator for Thin-Film Solar Cells." Solar Energy 13:149. Larson, D. C. 1980. "Optimization of Flat Plate Collector—Flat Mirror System." Solar Energy 24:203. McDaniels, D. K. et al. 1975. "Enhanced Solar Energy Collection Using Reflector Solar-Thermal Collector Combinations." Solar Energy 17:277. Mclntire, W. R. 1979. "Truncation of Nonimaging Cusp Concentrators." Solar Energy 23:35. Mclntire, W. R. 1980a. "Stationary Concentrators for Tubular Evacuated Receivers: Optimization and Comparison of Reflector Designs." Proc. 1980 Ann. Meeting, 505. AS Int. Solar Energy Society. Mclntire, W. R. 1980b. "Elimination of the Optical Losses Due to Gaps Between Absorbers and Their Reflectors." Proc. 1980 Ann. Meeting 3.1:600. AS Int. Solar Energy Society. Mclntire, W. R. 1981. "New Reflector Design Which Avoids Losses Through Gaps Between Tubular Absorbers and Reflectors." Solar Energy 25:215. Mills, D. R. 1978. "The Place of Extreme Asymmetrical Non-Focusing Concentrators in Solar Energy Utilization." Solar Energy 21:431. Mills, D. R. and Giutronich, J. E. 1978. "Ideal Prism Solar Concentrators." Solar Energy 21:423. Mills, D. R. 1981. "Two-Stage Tilting Solar Concentrators." Solar Energy 25:505. O'Gallagher, J. J., Rabl, A., Winston, R. and Mclntire, W. 1980. "Absorption Enhancement in Solar Collectors by Multiple Reflections." Solar Energy 24:323. Ploke, M. 1967. "Lichtfuehrungseinrichtungen mit starker Konzentrationswirkung." Optik 25:31. Rabl, A. 1976a. "Comparison of Solar Concentrators." Solar Energy 18:93. Rabl, A. 1976b. "Optical and Thermal Properties of Compound Parabolic Concentrators." Solar Energy 18:497. Rabl, A. 1976c. "Solar Concentrators with Maximal Concentration for Cylindrical Absorbers." Appl. Opt. 15:1871. Rabl, A. and Winston, R. 1976. "Ideal Concentrators for Finite Sources and Restricted Exit Angles." Appl. Opt. 15:2880. Rabl, A. 1977. "Radiation Transfer Through Specular Passages." Int. J. Heat Mass Transfer 20:323. Rabl, A., Goodman, N. B., and Winston, R. 1979. "Practical Design Considerations for CPC Solar Collectors." Solar Energy 22:373. Rabl, A., O'Gallagher, J., and Winston, R. 1980. "Design and Test of Nonevacuated Solar Collectors with Compound Parabolic Concentrators." Solar Energy 25:335. Reed, K. 1976. "Instrumentation for Measuring Direct and Diffuse Insolation in Testing Thermal Collectors," In Optics in Solar Energy Utilization II, Vol. 85. SPIE. Seitel, S. C. 1975. "Collector Performance Enhancement with Flat Reflectors." Solar Energy 17:291. Selcuk, M. K. 1979. "Analysis, Development and Testing of a Fixed Tilt Solar Collector Employing Reversible Vee-trough Reflectors and Vacuum Tube Receivers." Solar Energy 22:413. Tabor, H. 1958. "Stationary Mirror Systems for Solar Collectors." Solar Energy 2:27.
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Tabor, H. 1966. "Mirror Boosters for Solar Collectors." Solar Energy 10:111. Welford, W. T. and Winston, R. 1978. The Optics of Nonimaging Concentrators. New York: Academic Press. Window, B. and Bassett, I. M. 1981. "Optical Collection Efficiencies of Tubular Solar Collectors with Specular Reflectors." Solar Energy 26:341. Winston, R. 1974. "Solar Concentrators of a Novel Design." Solar Energy 16:89. Winston, R. 1976. "Dielectric Compound Parabolic Concentrators." Appl. Opt. 15:291. Winston, R. and Hinterberger, H. 1975. "Principles of Cylindrical Concentrators for Solar Energy." Solar Energy 17:255. Winston, R. 1980. "Cavity Enhancement by Controlled Directional Scattering." Appl. Opt. 19:195. Winston, R. and Welford, W. T. 1979. "Geometrical Vector Flux and Some New Nonimaging Concentrators." J. Opt. Soc. Amer. 69:532.
7. TRACKING CONCENTRATOR TYPES
In principle any collector, even a flate plate, could be made to track the sun, but in practice one will usually resort to tracking only with collectors of sufficiently high concentration ratio; say, greater than about 5. The collector types that are most suitable for tracking will be surveyed in this chapter. Since even direct solar radiation is not perfectly collimated but comes from a range of directions (which is further enlarged by optical errors), the receiver must have a minimum size if it is to intercept most of the incident radiation. The choice of the optimal absorber size involves a compromise between optical and thermal performance. If the absorber is too large most of the incident solar radiation will be intercepted but the heat losses are excessive. On the other hand, a very small absorber has low heat losses but it will also miss too much of the available solar radiation. The problem of choosing the receiver size (i.e., the concentration ratio) is illustrated in Section 7.1 for the important case of parabolic reflectors. However, the sizing rules of Section 7.1 are very approximate and do not account accurately for the statistical distribution of optical errors and off-axis aberrations. Readers who need a more accurate optical analysis of parabolic reflectors are referred to Chapter 8. 7.1 PARABOLIC REFLECTORS One of the best-known solar concentrators is the parabolic reflector; it can be built either as a trough or as a dish. The absorber can take a variety of shapes, the most common being flat or round. The parabola is the unique reflector shape that focuses a collimated beam of radiation into a single point. Perfect focusing is possible only for rays that are incident parallel to the optical axis (equals symmetry axis) of the parabola. A collimated beam coming from other directions will not only miss the focus but due to off-axis aberrations of the parabola it will not even converge into a single point. If the absorber is to intercept all rays, it must be made sufficiently large. The example of the parabolic trough reflector with cylindrical absorber tube in Fig. 7.1.1 serves to illustrate this procedure for focusing collectors. The absorber tube is placed concentrically around the focal line. If the ray with the largest deviation 6a is to reach the absorber just barely, as shown by the 170
Tracking Concentrator Types
171
dashed line in Fig. 7.1.1, then the concentration must be
where 4> is the rim angle < AOB. The rim angle is related to aperture width D = 2xA and focal length /by
The maximum of Eq. (7.1.1) occurs at <j> = 90° and falls a factor ir short of the ideal limit. This is typical of all single stage focusing concentrators; i.e., they reach only one-fourth to one-half of the thermodynamic concentration limit. Analogous results for other geometries of parabolic reflectors are listed in Table 7.1.1. The failure to reach the thermodynamic limit is related to the angular acceptance characteristic (See Fig. 6.1.2). Single parabolas do not have a sharp angular cutoff but accept some rays at angles beyond the nominal acceptance angle 9a (which has been denned as the largest incidence angle for which all or almost all rays on the aperture reach the receiver). Actually this simple analysis is not the complete answer since the distribution of incident rays does not have a sharp cutoff; also some rays from outside the nominal acceptance angle are accepted. An accurate analysis is quite involved. As a simple shortcut one frequently takes the rule of choosing Ba equal to twice the rms width
Figure 7.1.1 Absorber radius a and acceptance halfangle 6a for a focusing parabola.
TABLE 7.1.1 Concentration Ratio C and Mirror Surface/Aperture Area Ratio Am/Aat> as Function of Rim Angle 0 and Acceptance Half-Angle 8a for Parabolic Reflectors
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173
7.2 FRESNEL REFLECTORS 7.2.1 Line-focus and point-focus central receivers The smooth optical surface of a reflector or lens can be broken into segments, a device invented by Fresnel. An example of a point-focus Fresnel mirror system is the central receiver in Fig. 1.3.11. Fresnel reflectors can be very useful for solar energy, especially in large installations. As one increases the size of parabolic collectors, the mechanical problems of weight and wind loading increase. In practice it appears to be uneconomical to build parabolic collectors with aperture areas much larger than 100 m2. When one needs larger areas, one would therefore have to install several collector modules, each with its own receiver. As discussed in Section 12.4, collecting the heat from the individual receivers to a central point of use can be quite expensive. Therefore it may be better to collect the solar energy optically by means of a central receiver. The receiver is mounted on top of a tower, surrounded by a field of so-called heliostats, mirrors that track the sun individually as shown in Fig. 1.3.11. Linear Fresnel mirror systems can also be built, with linear receiver and one-axis tracking reflectors (see Fig. 1.3.8). The optical analysis of Fresnel reflectors is fairly complicated. Since the location of the reflectors is fixed,1 the angle of incidence of the sun on the reflector varies with time. This causes shading and blocking. Shading occurs if direct sunlight fails to reach a mirror because the mirror is in the shadow of another mirror; blocking occurs if light reflected by a mirror fails to reach the absorber because it is intercepted by the backside of another mirror. In the interest of efficient mirror utilization, the reflector segments should be spaced far enough apart to avoid excessive shading and blocking. The optimal spacing depends on average incidence angles and on the relative cost of reflector area and the rest of the system. The ratio of heliostat area over total ground area is called ground cover factor ^. Values of \f/ between 0.2-0.6 are practical for Fresnel reflectors that track around the north-south axis or about two axes. For linear Fresnel reflectors with east-west tracking axes the shading and blocking effects are less severe, and \^ can be larger. There is a further complication caused by varying incidence angles of the sun. The size and shape of the solar "images" from the individual reflectors varies with time of day and year. This effect is particularly complicated if the individual mirrors have some curvature. If the number of mirrors is large, the gain in concentration achievable by curving the reflectors is small. When the number of mirrors exceeds on the order of 100 for linear-focus or 10,000 for point-focus Fresnel systems, one gets diminishing returns from the extra expense2 of curving the mirrors, and one may be better off with flat 'Usually the reflectors are mounted on the ground (e.g., for the central receiver) or in a tilted plane (e.g., for small line-focus Fresnel collectors). 2 The cost of curving mirrors varies greatly with manufacturing technique. It is easy to impart cylindrical curvature to glass mirrors. Spherical curvature in glass mirrors is more difficult and causes stress. One technique that produces low-cost reflectors of nearly spherical curvature is to stretch a reflective plastic film over a circular frame and apply negative pressure.
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mirrors. On the other hand, when the number of mirrors is small, one can increase the concentration significantly by curving the mirrors. An important example, spherically curved heliostats for the point-focus central receiver, is analyzed in the following section. In view of the complexity of the problem, it is not surprising that no closed form analytical formulas are available for calculating the optical performance of central receivers. Only some partial solutions have been attempted. For example, Riaz [1976] has derived equations for the effects of shading and blocking. Various fast algorithms have been developed for carrying out the computations [Vant-Hull, 1976] but the complete analysis seems to require a computer program. Several computer programs have been developed for this purpose. The HELIOS program [Biggs and Vittitoe, 1979] is a very accurate and flexible program that can model such details as circumsolar radiation, arbitrary reflector shapes, and second stage concentrators; however, a program like HELIOS is slow and expensive to run. Other examples of computer programs for central receivers are MIRVAL Leary and Hankins, 1977], DELSOL [Dellin and Fish, 1979], and SCRAM [Bergeron and Chiang, 1980]. The latter make various approximations and are much faster than HELIOS, but they are less accurate and less (or far less) flexible. For more analytical approaches to calculating the optical behavior of central receiver systems the reader is referred to the work of Riaz [1976] and of Vant-Hull and his co-workers [Walzel et al., 1977; Lipps et al., 1978; and Vant-Hull, 1976]. A particularly interesting result of these calculations is shown in Tables 7.2.1a and 7.2.1b [Eicker, 1979]. These two tables show the field efficiency (essentially equal to the incidence angle modifier) K(6Z, 4>s) for particular central receiver designs. For any values of the solar zenith angle 6Z and the solar azimuth angle $, the product of beam irradiance and K(6Z, s) yields the radiation intercepted by the receiver. The field efficiency as listed in Tables 7.2. la and 7.2.1b includes (and is limited to) the effects of incidence angle cosine, shading, blocking, mirror reflectivity, tower shadowing, atmospheric attenuation between mirror and receiver, and intercept factor (i.e., loss of rays that miss the receiver). The atmospheric attenuation depends on the distance from mirror to receiver, and is approximately 10% per kilometer. (For an accurate model of this attenuation the reader is referred to Pitman and Vant-Hull [1982].) The reflectivity of the mirrors is assumed to be p = 0.90. The absorptivity of the receiver is not included in these tables. The need for two different tables arises from the difference in mirror field design for small and for large systems. For small systems at intermediate latitudes the heliostats will be deployed only to the north of the tower, whereas in large systems the mirror field will surround the tower (albeit with more mirrors to the north than to the south); this design difference reflects different economies of scale in tower and heliostat costs. Table 7.2. la is for a small field of 2.4 MW, capacity, while Table 7.2.1b is for a large field of 195 MW, capacity. Both are optimized for a geographic latitude of 35° and for a specified ratio of tower and heliostat costs. The reader should understand that
175
Tracking Concentrator Types TABLE 7.2.1
Field Efficiency pK(Oz, $,) for Two Central Receiver Designs.8
7.2. la Small Central Receiver: 2.4 MW, s = azimuth (degrees, south = 0) Elevation = 90° - 6, (degrees, horizon = 0°)
5 15 25 45 65 89.5
0
30
60
75
90
110
0.384
0.404 0.687
0.366 0.576 0.662 0.757 0.754 0.748
0.330 0.495 0.584 0.708 0.753 0.726
0.300 0.429
0.240 0.367 0.445 0.603 0.689 0.736
0.701 0.789
0.814 0.811 0.723
0.771 0.814 0.806 0.729
0.521 0.661 0.724 0.730
130 0.212 0.315 0.391 0.544 0.642 0.736
7.2.1b Large Central Receiver: 195 MW, <£j = azimuth (degrees, south = 0) Elevation = 90° - 02 (degrees, horizon = 0°)
5 15 25 45 65
89.5
0
30
60
0.216 0.446 0.560 0.719 0.684 0.683
0.215 0.448 0.558 0.640 0.670 0.683
75
90
0.206 0.425 0.537 0.626 0.671 0.686
0.204 0.423 0.522 0.618 0.668 0.672
0.199 0.405 0.516 0.605 0.660 0.682
110
0.194 0.392 0.498 0.594 0.655 0.687
130
0.192 0.385 0.491 0.599 0.641 0.681
"Includes losses due to cosine, tower shadowing, blocking and shading, atmospheric attenuation, intercept factor, and a value p = 0.90 for the heliostat reflectivity (from Eicker [1979]).
different designs, as appropriate for different latitudes, cost ratios or system sizes, may have quite different incidence angle modifiers. A 5-MW, central receiver test facility has been operated in Albuquerque since 1978, and a 10 MWe central receiver electric power plant (using a steam turbine) in Barstow, California, has been producing power since 1982. Measurements at the Albuquerque test facility have confirmed the calculations of the HELIOS computer program, and optical errors as small as a few mrad have been achieved [Biggs and Vittitoe, 1979].3 7.2.2
Tracking motion for linear fresnel reflectors
Figure 7.2.1 shows a line-focus Fresnel reflector system with flat reflectors in cross section. As discussed in Chapter 5, a linear reflector can be analyzed by projecting all rays onto the plane normal to the reflectors. Therefore the ray trace diagrams in Fig. 7.2.1 can be interpreted as a projected ray diagram, which is independent of the elevation of the rays from the plane of the paper. Let us calculate the tracking angle $, of the zth reflector, defined as angle 3 More information on central receivers can be found in Battleson [1981], in Hildebrandt et al. [1980], and in ASME [1984].
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Figure 7.2.1 Tracking motion for line-focus Fresnel concentrator, (a) Sun overhead, (b) Sun at incidence angle 8. \f/, is tracking angle of z'th reflector.
between the rth reflector and the normal of the plane in which the reflectors are mounted. Consider first the case where the sun is overhead; i.e., when the incidence angle relative to the aperture normal is 0 = 0 as in Fig. 7.2.la. Then it is obvious from Snell's law that
where , is the angle between optical axis and the line from focus to reflector. If the sun moves now by an angle 6, then Snell's law requires each mirror to
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111
rotate by an angle 6/2 as shown in Fig. 7.2.1b. Hence the tracking angle is
and all mirrors turn by the same angle to follow the sun. This kind of tracking motion can be accomplished by simple mechanical linkage. With pointfocus Fresnel reflectors the tracking motion can also be achieved by mechanical linkage to a central drive, as demonstrated with the solar furnaces built in Genoa, Italy, and in Atlanta, GA, according to a design by Francia[1968]. 7.2.3
Off-axis aberrations of spherical reflectors
In order to find the optimal curvature of heliostats for a central receiver it is instructive to examine the off-axis aberrations of spherical reflectors. Even if the actual surface of a heliostat is not exactly spherical, the curvature will usually be so slight that one can treat the heliostat as a spherical segment whose radius of curvature is large compared to its width. Consider a collimated beam of rays that is incident on a spherical reflector at an angle n from the normal C through the center of the reflector. It is helpful to single out two sets of rays from this beam: the sagittal ray fan and the tangential ray fan. The tangential ray fan consists of the rays that lie in the plane spanned by the reflector normal C and by the incident direction. The sagittal ray fan contains the rays that lie in the plane perpendicular to the tangential ray fan. The focal points for the tangential and the sagittal ray fans are different and coincide only at normal incidence. Let us calculate the tangential and sagittal focal lengths as a function of incidence angle n. Figure 7.2.2 shows two rays from the tangential fan. One ray R{ is incident at the center of the reflector where the reflector normal is C. The other ray ^2 hits the reflector a small distance y = d away from the center, where the
Figure 7.2.2 Two rays from the tangential ray fan (Adapted from Biggs and Vittitoe[1979]).
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Active Solar Collectors and Their Applications
reflector normal D makes a small angle ft with the normal C. The distance d is, of course, assumed to be small compared to the radius of curvature. Since the radius of curvature is so large, the small upward (i.e., in the z direction) displacement of the reflector surface at y = d has been neglected. In order to find the focal length ft of the tangential ray fan, we note that after reflection the rays Rt and R2 intersect at F,, making an angle 2/3. The law of sines applied to the triangle OFF, yields
or
The relation between d and /3 is readily found from the fact that C and D point towards the center of a sphere of radius r. Since /3 is small, we can write
The smallness of /? also allows us to approximate cos (2/3 — /u) by cos n. With this approximation Eqs. (7.2.4) and (7.2.5) can be combined to yield
One recognizes r/2 as the focal length f0 of a spherical reflector at normal incidence. Hence the focal length for the tangential fan shrinks with incidence angle /* like cos n. (Strictly speaking the /? dependence of f, in Eq. (7.2.4) means that there is no perfect focus at F,; however, the variation of / with /3 is small compared to r and can therefore be neglected.) To find the sagittal focal length^ consider Fig. 7.2.3. Ray R, hits the center of the reflector at O, as before. But ray R2 is in the sagittal fan and hits the reflector at a point P, at a small distance d along the x axis. The incident direction i of the ray R2 is
in the (x, y, z) coordinate system of Fig. 7.2.3. The reflector normal at point Pis
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Figure 7.2.3 Two rays from the sagittal ray fan (Adapted from Biggs and Vittitoe [1979]).
since /? « 1. By Eq. (5.1.4) the ray R2 is reflected in the direction
for small 0. In terms of the components of f the angle 7 at which the rays intersect after reflection is given by
since the triangle OPFS has a right angle at O. The angle 7 is related to the focal length fs by
For the relation between /3, d, and radius of curvature r we still have Eq. (7.2.5). Combining Eqs. (7.2.5), (7.2.10), and (7.2.11) we obtain the sagittal focal length f;.
To sum it up let us designate the focal length at normal incidence byf0; it is half the radius r.
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Active Solar Collectors and Their Applications
Then the focal length fs and f, of the sagittal and tangential ray fans at an incidence angle /j, from the normal of the reflector center are given by
In order to summarize these results, we show the reflected portions of both the tangential ray fan and the sagittal ray fan in Fig. 7.2.4. The collimated incident beam (not shown) comes from an off-axis (M ¥= 0) direction. The tangential ray fan is roughly vertical, whereas the sagittal ray fan is roughly horizontal in the perspective drawing of the figure. If the reflector were rotated toward a zero angle of incidence, the two focal lines would become shorter and approach each other until at n = 0 they coalesce into a single focal point O at a distance f0 from the center of the spherical reflector. 7.2.4 Curvature of heliostats for central receiver The results of the preceding section have interesting implications for the choice of curvature of heliostats. This is illustrated by means of Fig. 7.2.5. The distance from heliostat to receiver is called slant range and is designated by D. The heliostat is assumed to be round with diameter w. The top of the figure shows the tangential ray fan; the bottom shows the sagittal ray fan, corresponding to a collimated beam that is incident on the heliostat at an angle n. The tangential ray fan is brought to focus at f, and diverges to a width h, by the time it strikes the receiver. The sagittal ray fan converges to Fs and has a width hs, where it crosses the receiver plane. The impact point of all other rays on the receiver will be within an ellipse of major and minor diameter hs and h,. From the geometry of Fig. 7.2.5 it follows that
and
Figure 7.2.4 Reflected tangential ray fan and sagittal ray fan and the corresponding focal lines for a collimated beam incident off-axis on a spherical reflector (From Biggs and Vittitoe[1979]).
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18-1
Figure 7.2.5 Tangential ray fan (top) and sagittal ray fan (bottom) for spherical heliostat at a distance D from the receiver (Adapted from Biggs and Vittitoe [1979]).
The area of the image is irhji, and varies with the focal length f0 of the heliostat. Combining Eqs. (7.2.14) through (7.2.17) and taking the derivative of hji, with respect tof0, one can show that the image area is minimized if the focal length is chosen as
In particular, at zero incidence angle n = 0 the optimal focal length equals the slant range D:
For small and intermediate incidence angles the right-hand side of Eq. (7.2.18) varies only slowly with \t, and hence the choice ^o — D is an excellent compromise for the optimal focal length of a heliostat. It places the "circle of least confusion" at the receiver and it makes hs and h, equal:
If the incident radiation is not collimated but comes from the sun with angular radius As, then the solar image at the receiver has a diameter
For flat heliostats the image diameter would be
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Active Solar Collectors and Their Applications
If the incidence angle n is less than 60°, focused heliostats with Jo = D produce a smaller image than flat heliostats. Of course, this effect is significant only if w is sufficiently large compared to 7.3 FRESNEL LENSES Use of an ordinary lens, Fig. l.S.lOa, is impractical in most solar applications because the lens would have to be too thick. The mass per aperture ratio is proportional to aperture width, and for widths larger than a few centimeters the mass and weight become excessive. As an alternative, one can use a Fresnel lens. The optical performance of the Fresnel lens, Fig. 1.3.10b, is almost equivalent4 to an ordinary lens, Fig. 1.3.10a. In principle both sides of a Fresnel lens could be grooved, but in practice one will use lenses with one smooth surface. Furthermore, the grooves should face down to minimize dirt accumulation. Flat Fresnel lenses with smooth surface towards the sun have poor transmission because of large incidence angles at the outer prisms. Also they have large off-axis aberrations. Kritchman, Friesem, and Yekutieli [1979] found an elegant lens design that avoids these problems, and that comes very close to the thermodynamic limit of concentration
for an acceptance half-angle 6a. Their starting point is the extreme ray principle for ideal concentrators. The collector is assumed to be symmetrical with specified acceptance half-angle 0a, and the absorber is taken to be flat. According to the extreme ray principle, all rays incident at + 6a (— 6a) must reach the left (right) edge of the absorber in Fig. 7.3.1. In other words, the lens must have two perfectly sharp off-axis focal points, one for rays with + 8a and one for rays with — 0a. If this condition is satisfied, then all rays with \&\ < 6a will reach the absorber. Mathematically this requirement imposes two equations for the slopes of each prism facet. These equations cannot be satisfied if the contour of the smooth front surface is specified because then only one parameter, the back slope, can be chosen freely. There is, however, a solution if one allows the contour also to be determined by the extreme ray condition. The solution does not have a closed form, but can be calculated by computer. Figure 7.3.1 is actually an example of such a lens with an acceptance half-angle 6a = 10°. Included in this figure are the ray paths for two incident directions, 6 = 0° and 6 = 10°; of course, the rays coming from 0 = 6a = 10° all hit the edge of the absorber. The geometrical 4
The principal difference lies in the fact that the optical path length for different rays is different in a Fresnel lens whereas in an ordinary lens the optical path from object to image is the same for all rays. This difference is inconsequential for solar energy since imaging and coherence properties are irrelevant.
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Figure 7.3.1 Shape of Fresnel lens with maximal concentration for acceptance half-angle 10° (From Kritchman et al. [1979b]). (a) Incidence angle 10°. (b) Incidence angle 0°.
concentration of these lenses is at least 75% of the thermodynamic limit.5 Like the CPC, they can be truncated with little loss in concentration. That is, the outer portions of the lens do not intercept much radiation but require a large amount of material per aperture area. In practice it will probably be reasonable to truncate the lens to about half of the full depth; for the lens shown in Fig. 7.3.1 such a truncation reduces the concentration from 0.75/ sin 10° to 0.67/sin 10°. If designed with an acceptance angle equal to the solar size (i.e., Oa = A, = 5 mrad), such a lens reaches a concentration ratio of 0.745/sin Ba = 149 in line-focus geometry. Truncated to half of its full depth its concentration is still equal to 133. Compared to other Fresnel lenses the design of Kritchman, Friesem, and Yekutieli has the following advantages: (i) it reaches the highest geometric concentration possible with a Fresnel lens (ii) reflection losses are small because the angle between incident ray and first surface is close to the angle between exiting ray and prism surface (iii) the concave shape adds mechanical strength to the lens 5 It is easy to see why this lens does not quite reach the thermodynamic limit despite the extreme ray principle used for its design. Some of the rays emitted from the absorber to the lens hit the wrong prism facets and are refracted to leave the lens outside the acceptance angle. In general a Fresnel lens cannot reach the thermodynamic limit because of the discontinuity in optical path from one prism facet to another.
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(iv) the loss of performance for nonzero elevation angles (i.e., when sun is not in the plane of the paper of Fig. 7.3.1) is less severe than for other linear Fresnel lenses; by the same token it is less sensitive to chromatic aberations The same lens design can be used for point-focus systems, by rotating the contour of the line-focus lens about the optical axis. Several other Fresnel lens designs have been published recently that come fairly close in performance to the optimal design of Kritchman, Friesem, and Yekutieli. Collares-Pereira [1979] apprixmated the curved lens profile by two straight portions and used a second stage CPC reflector in the focal zone. O'Neill [1979] took as starting point the criterion for minimal reflection loss: equal angles between light rays both at front and back of the lens. Together with the requirement for a smooth front surface this determines the lens uniquely. Since the design of Kritchman, Friesem, and Yekutieli yields almost equal angles at front and back between light rays and prism surfaces, it is not surprising that O'Neill's design approaches the design for maximal geometric concentration very closely. Compared to reflective optics, Fresnel lenses offer several advantages: (i) A Fresnel lens has dramatically greater tolerance to surface contour errors as shown by O'Neill [1979].6 (ii) A Fresnel lens can do double duty as collector cover, protecting the inside not only from heat loss by forced convection, but also from dirt. In a well sealed collector light rays have to penetrate only once through a layer of dirt. In an exposed reflector, on the other hand, light rays have to penetrate the dirt layer twice, first on the way to the reflecting surface and then again after reflection, because any reflector for outdoor exposure has a sufficiently thick protective coating in front of the aluminum or silver surface itself that incident and reflected rays will be affected by different dust particles. Despite these advantages Fresnel lenses do have problems of their own. Above all, the focal length of linear Fresnel lenses shrinks as the elevation of the incident ray from the plane of the paper increases. This is illustrated schematically in Fig. 7.3.2a. A numerical example for a particular lens is shown in Fig. 7.3.2b. This effect is so severe that Fresnel lenses with horizontal east-west tracking axis are impractical. Even with polar mount, which minimizes the elevation changes to ± 23.5°, use of a Fresnel lens is marginal. In fact most linear Fresnel lenses that have been built have been intended for installations with seasonal adjustment of the tracking axis or even two-axis tracking. If the tracking axis of a linear Fresnel lens collector 6 This can be made plausible by considering rays passing through the center of the lens. If the slope of the facet differs from the design direction by an angle S, then the refracted ray differs from its design direction by (n — 1) S, where n is the index of refraction. In a reflector, by contrast, a contour error S causes a deviation 25 of the reflected ray. For typical lens materials the index of refraction is n s 1.5 and the sensitivity to contour errors near the center of the lens is a factor 4 smaller than for reflectors. (The sensitivity to tracking errors is, of course, the same as for reflectors because a tracking error S is equivalent to an enlargement of the source by a.)
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Figure 7.3.2 (a) Linear Fresnel lens, incident beam in plane of paper (solid line, focus F), and out of plane (dashed line, focus F'). (b) Focal length/ of particular linear Fresnel lens for nonplanar incidence (0, = elevation from plane of paper) (From CollaresPerieraetal. [1977]).
is to be fixed, a geometric concentration ratio of about 15 is the upper limit [Collares-Pereira, 1979; Kritchman, Friesem, and Yekutieli, 1980]. Ultimately the choice between reflective and refractive concentrators will be made in terms of cost per delivered energy. This will depend critically upon cost and durability of mirror and lens materials. Glass is very durable but as lens material it is ill-suited at least for the grooved surface because the high surface tension and the high softening temperature of glass prevent accurate molding of small prisms. Plastics, especially acrylic, are easy to mold, but their durability is questionable, the main problem being resistance to scratching and to UV degradation. As for the optical transmission through Fresnel lenses, one can expect the following losses for a well
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Active Solar Collectors and Their Applications
designed lens made of acrylic (index of refraction is n = 1.491) [O'Neill, 1979]: Loss type
Fraction of incident radiation lost
Front surface reflection Absorption and scattering in acrylic Back surface reflection
4.3% 2.0% 4.0%
In addition there may be losses up to 5% from the intercept factor because the receiver will usually be made small enough that some rays miss the receiver. 7.4
FIXED REFLECTORS WITH TRACKING RECEIVERS
As an alternative to moving a large reflector, one can design systems where the reflector is fixed and only a small receiver needs to track the sun. Three such systems are known: the hemispherical reflector, the circular cylindrical reflector, and the reflector slats on circular cylindrical mount. 7.4.1 Spherical reflector Collimated radiation incident on a spherical reflector will, after one or several reflections, cross the line that extends through the center of the sphere in the direction of the incident direction. Rays close to this line reach the focus, which is on this line, halfway between reflector and center of the sphere, as shown in Fig. 7.4. la. Other rays intersect this line between the focus F and the apex A. Therefore a receiver that extends from focus F to apex A will intercept all rays. Because of the spherical symmetry of the reflector, only the receiver needs to track if the radiation source moves. As a solar collector such a system can attain geometric concentration ratios up to about 270 [Clausing, 1976]. The flux concentration along the receiver is quite nonuniform [Kreider, 1975]. Even though the focusing is independent of incidence angle, the effective aperture is not. This is illustrated in Fig. 7.4.1b, where the reflector portion from R to S is inactive for the incidence angle shown. 7.4.2 Circular cylindrical reflector with tracking receiver One could use the design principle of the hemispherical reflector for a twodimensional arrangement as well; i.e., a tracking receiver plate placed along a circular cylindrical reflector trough. However, the concentration ratio of such an arrangement is less than 2 and too low to be practical. However, as the ray trace diagram of Fig. 7.4.2 shows, a sufficienty large tubular absorber placed slightly below the focal line (i.e., the line halfway betwen apex and
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Figure 7.4.1 Fixed spherical reflector with tracking receiver, (a) Normal incidence, (b) Cosine loss at off-normal incidence (Adapted from Clausing [1976]).
center of the reflector) can intercept most of the reflected rays. This arrangement was investigated by Tabor and Zeimer [1962], who designed and tested a concentrator with geometric concentration ratio 2.5 and a fairly large acceptance angle 28a = 17°. More recently, a group at Lawrence Livermore Laboratory [Gerich, 1979] has built and tested a similar collector type with inflated cylindrical reflector that needs approximately 12 tilt adjustments per year and that is expected to be suitable for temperatures up to 175°C. An interesting variation of this concept has recently been designed and tested by Edmonds et al. [1983]. It uses a prismatic reflector [Mills and Giutronich, 1978] as second stage to boost the overall concentration ratio close to the thermodynamic limit; its main application appears to be for photovoltaic cells.
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Figure 7.4.2 Ray trace diagram for circular cylindrical reflector with cylindrical absorber. In this example the incident beam is 3° off-axis (= line absorber center to reflector center) (From Gerich [1978]).
7.4.3 Reflector slats on circular cylindrical mount The third-fixed reflector scheme consists of narrow flat reflector slats that are mounted on a circular cylindrical surface. The cross section is shown in Fig. 7.4.3. If the mirror slats have the correct slope, this system will always produce a perfect line focus (in the limit of infinitely narrow slats) regardless of incidence angle. The focus is on the same circle as the reflector slats. Figure 7.4.3 explains the basic geometry. The solar radiation as shown happens to be incident at an angle of 30° from the vertical, and the rim angles 75° to the north and 45° to the south of vertical (for east-west through axis) are approximately optimal for deployment at 35° latitude. A particular ray is shown incident on a mirror facet located at an angle /3 from the line to the sun. Simple geometry determines the mirror facet angle at this point
189
Figure 7.4.3 Fixed mirror slats on cylindrical mount with tracking receiver. Basic geometry (Adapted from Eggersetal. [1979]).
to be (8/4 below the horizontal. The angles of the other facets are determined in the same way. Now consider what happens as the sun moves by an angle 5 to the right. Then all reflected rays move by an angle 5 to the left. The included angle between two rays reflected from two different facets therefore remains constant. According to a well known theorem7 the focus therefore moves on the same circle on which the mirror facets are positioned. In the limit of infinitely narrow slats the focus is perfect. There are, however, losses due to shading and blocking [See Russell, 1974; Eggers et al., 1979]. 7.5
CONCENTRATOR CONFIGURATIONS
FOR LOW-COST MANUFACTURE
Some manufacturing processes naturally lend themselves to the reproduction or approximation of certain reflector geometries. Flat float glass mirrors, for example, can be used to approximate the profile of a CPC. Reflector surfaces of near spherical shape [e.g., Brantley, 1977] can be obtained by stretching a metallized plastic film over a circular frame and applying air pressure to one side. The profile is not exactly spherical because the fixed frame prevents the film from stretching uniformly in all directions. A very slight air pressure difference, less than 1% of atmospheric pressure, is adequate for maintaining the proper shape, and the surface is close enough to a true paraboloid to yield concentration values of several hundred with 2axis tracking. Air-inflated reflectors may also be suited for some of the fixed reflector systems described in the previous subsection; e.g., see Tabor and Zeimer [1962] and Gerich [1978]. The catenary or hyperbolic cosine profile can be obtained by suspending 7 For three points A, B, and C on a circle the angle < ABC remains constant if the end points A and C are fixed while the apex B moves along the circle.
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Active Solar Collectors and Their Applications
metallized plastic film from a frame. A parabolic trough reflector can also be approximated by applying an appropriate bending moment to the edges of an elastic reflector sheet [McCormick, 1981]. An intriguing technique for producing a low-cost point-focus concentrator has been invented by Steenblick at Georgia Institute of Technology [Walton, 1980]. A properly designed spiral pattern is cut from a flat sheet of reflector material. By slightly winding the spiral like a spring and mounting it on a planar surface with appropriate support points and tilt angles, one obtains a point-focus reflector. A full-scale model with concentration ratio 500 has been successfully demonstrated. This is really a Fresnel reflector approximation of a parabolic dish. As an alternative manufacturing technique for point-focus Fresnel reflectors one can press a fine circular pattern of grooves into a transparent plastic sheet and cover it with a reflective coating. The possibility of mounting a point-focus reflector on a flat support structure is attractive because one can take advantage of the mass production potential of heliostats for central receiver systems and use the basic structure of a heliostat to make individual point-focus collectors that are much smaller than a central receiver system. A particularly interesting design encloses the concentrator inside an inflated transparent bubble. This approach has been investigated both by Boeing and by General Electric. It is suitable for heliostats as well as for parabolic dishes. This approach has the main advantage of eliminating the wind load on the concentrator itself, therefore permitting light weight, lowcost fabrication. A heliostat inside a bubble could for instance be made by stretching an aluminized plastic film over a circular frame. Since the forces on an inflated bubble become excessive if it is made larger than several tens of meters in diameter, it is not practical to enclose an entire central receiver plant. Hence only individual heliostats will be enclosed and light has to traverse the enclosures twice, at fairly high optical loss. The main difficulty of the bubble approach seems to lie in the stringent requirements for the bubble material. The material must have a high specular transmittance, it must have sufficient mechanical strength, and it must have a long life when exposed to sunlight—all that at very low cost. Straight reflector cones can be made quite easily by appropriate bending of a flat reflector sheet. The absorber is a narrow cone extending along the symmetry axis. Despite the need for 2-axis tracking the achievable concentration is relatively low, in the range of a few hundred [Kurzweg, 1980]. 7.6 SECOND STAGE CONCENTRATORS In collectors with rim angle less than 90° the radiation incident on the absorber has a limited angular spread and can therefore be further concentrated. To illustrate the design of second stage concentrators, we consider the line-focus parabola in Fig. 7.6.1. Without a second stage the receiver would be the flat surface from B to B'. If the acceptance half-angle is 6a and no
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191
Figure 7.6.1 Second stage CPC concentrator for focusing parabola. Absorber = CC.
radiation is to miss the receiver, the concentration AA'/BB' is by Table 7.1.1
where 4> is the rim angle. The — 1 accounts for the shading of the aperture by the receiver. Since the present discussion is most relevant for systems with high concentration, we can assume that 8a « 1 and neglect the — 1 on the right-hand side. A second stage CPC with acceptance half-angle, <j> boosts the concentration by a factor I/sin 0 to a value
for the system as a whole. For small rim angles the concentration of the two stage system approaches the thermodynamic limit, and even for $ as large as 30° it is still within 15%. For large rim angles the second stage becomes less effective. In a central receiver the rim angles may be fairly large, up to about 65°, and a single second stage CPC could not accomplish much; in that case one could, however, divide the heliostat field and the receiver into several separate sections each of which has in effect a small rim angle. As for the precise design of the second stage, it should be based on the design of Fig. 6.1.6 because the source of radiation (i.e., the first stage) is a finite distance away. For a flat absorber the ideal shape turns out to be compound elliptic rather than compound parabolic. An intriguing alternative second stage concentrator, called "trumpet"
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Active Solar Collectors and Their Applications
Figure 7.6.2 Second stage trumpet concentrator for focusing parabola. Absorber = CO. The crosses designate the same positions as the points B and B' in Fig. 7.6.1.
because of its shape, has recently been discovered by Winston and Welford [1979] and evaluated in great detail by Kritchman [1981a, 1981b]. For a flat absorber the trumpet has hyperbolic profile as shown in Fig. 7.6.2. The full profile extends all the way to the primary concentrator, and for a parabolic primary reflector that would cause complete shading. In practice a trumpet would be highly truncated, as suggested by the difference between the dashed line (complete profile) and the solid line (actual profile). For pointfocus geometry the shading loss for a reasonably truncated trumpet is only a few percent of the total incident radiation, while the gains in concentration can be on the order of 2. In choosing between a trumpet and a CPC as second stage, one should consider many factors besides geometric concentration; e.g., average number of reflections and absorption loss, problems of cooling the second stage, rejection of skew rays, shading, flux distribution at aperture, size, and weight, etc. For a detailed analysis the reader is referred to the papers by Kritchman [1981a, 1981b, 1982]. Whether one wants to use a second stage concentrator at all depends also on many factors, in particular, on the relation between desired concentration ratio and optical accuracy. The fact that for a given acceptance angle the concentration reached with a second stage is 2-4 times as high as for a system without a second stage concentrator is an obvious advantage for the design of ultrahigh-temperature power plants (using power cycles such as high-temperature gas turbines, magnetohydrodynamics, or thermionic conversion). However, it can be just as important for solar collectors of low or intermediate temperature, because the acceptance half-angle 6, can be doubled for a specified concentration, thus allowing a very significant relaxation of the mirror accuracy. For example, a power tower with effective mirror and tracking error Am = A^ = 4.7 mrad in its heliostats and with CPC second stage achieves as high a concentration as a power tower without CPC but with perfect mirrors. Contour errors of the second stage CPC are in-
Tracking Concentrator Types
193
significant as long as they are small compared to Oa; this is a crucial difference between an imaging Cassegrain system and the nonimaging CPC system. REFERENCES ASME. 1984. The February 1984 issue of the ASME J. Solar Energy Engineering 106:22-103, is devoted to central receivers. Battleson, K. W. 1981. "Solar Power Tower Design Guide: Solar Thermal Central Receiver Power Systems. A Source of Electricity and/or Process Heat." Report DE-81-02-65-84 SAND 81-8005. Albuquerque, NM: Sandia National Laboratories. Bergeron, K. D. and Chiang, C. J. 1980. "SCRAM: A Fast Computational Model for the Optical Performance of Point Focus Solar Central Receiver Systems." Report SAND 80-0433. Albuquerque, NM: Sandia Laboratories. Biggs, F. and Vittitoe, C. N. 1979. "The Helios Model for the Optical Behavior of Reflecting Solar Concentrators." Report SAND 76-0347. Albuquerque, NM: Sandia Laboratories. Boeing. 1976. "Central Receiver Solar Thermal Power System." Boeing Report SAN-111-76-2. Boeing. Brantley, L. W., Jr. 1977. "A Pressure Stabilized Solar Collector." In ERDA Conference on Concentrating Solar Collectors. Atlanta, GA: Georgia Inst. of Tech. Clausing, A. M. 1976. "The Performance of a Stationary Reflector/Tracking Absorber Solar Concentrator." In ISES Solar Energy Conference, Vol. 2, p. 304. Winnipeg, Canada. Collares-Pereira, M. 1979. "High Temperature Solar Collector with Optimal Concentration non-Focussing Fresnel Lens with Secondary Concentrator." Solar Energy 23:409. Collares-Pereira, M., Rabl, A., and Winston, R. 1977. "Lens-mirror Combinations with Maximal Concentration." Appl. Opt. 16:2677. Dellin, T. A. and Fish, M. J. 1979. "A User's Manual for DELSOL—A Computer Code for Calculating the Optical Performance, Field Layout, and Optimal System Design for Solar Central Receiver Plants," Report SAND 79-8215. Albuquerque, NM: Sandia Laboratories. Edmonds, I. R., Cowling, I. R., Meara, L. A., and Wheeler, B. 1983. "The Design and Performance of Ideal Solar Concentrators Based on the Prism-Assisted Cylindrical Reflector." Solar Energy 30:537. Eggers, G. H. et al. 1979. Report GA-A14209 (Rev.) UC-62. General Atomic. Bicker, P. J. 1979. Sandia Livermore Laboratories. Letter of 18 April 1979 to J. Thornton, Solar Energy Research Institute. Francia, G. 1968. "Pilot Plants of Solar Steam Generating Stations." Solar Energy 12:51. Gerich, J. W. 1978. "An Inflated Cylindrical Solar Concentrator." In Proc. 1978 Annual Meeting Am. Section Int. Solar Energy Soc., p. 889. Denver, CO. Gerich, J. W. 1979. "A Nontracking Inflated Cylindrical Solar Concentrator." In International Solar Energy Society Congress. Atlanta, GA. Hildebrandt, A. F. and Dasgupta, S. 1980. "Survey of Power Tower Technology." ASME. J. Solar Energy Engineering 102:91.
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Kreider, J. F. 1975. "Thermal Performance Analysis of the Stationary Reflector/ Tracking Absorber (SRTA) Solar Concentrator." J. Heat Transfer 97:451. Kritchman, E. M., Friesem, A. A., and Yekutieli, G. 1979a. "Highly Concentrating Fresnel Lenses." Appl. Opt. 18:2688. Kritchman, E. M., Friesem, A. A., and Yekutieli, G. 1979b. "Efficient Lens for Solar Concentration." Solar Energy 22:119. Kritchman, E. 1981a. "Optimized Second Stage Concentrator, "^pp/. Opt. 20:2929. Kritchman, E. 1981b. "Nonimaging Second Stage Elements—a Brief Comparison." Appl. Opt. 20:3824. Kritchman, E. 1982. "CEC Second Stage Concentrator." Appl. Opt. 21:751. Kurzweg, U. H. 1980. "Maximized Solar Flux Concentration Achievable With Axicon Collectors." Solar Energy 25:221. Leary, P. and Hankins, J. 1977. "A User's Guide for MIRVAL Computer Code for Comparing Designs of Heliostat Receiver Optics for Central Receiver Solar Power Plants." Report SAND 77-8280. Li verm ore, CA: Sandia Laboratories. Lipps, F. W. and Vant-Hull, L. L. 1978. "A Cell-wise Method for the Optimization of Large Central Receiver Systems." Solar Energy 20:505. McCormick, P. G. 1981. "Optical Evaluation of Cylindrical Elastic Concentrators." Solar Energy 26:519. Miller, O. E., McLeod, J. H., and Sherwood, W. T. 1951. "Thin Sheet Plastic Fresnel Lenses of High Aperture." J. Opt. Soc. Am. 41:11. Mills, D. R. and Giutronich, J. E. 1978. "Ideal Prism Solar Concentrators." Solar Energy 21:423. Nelson, D. T., Evans, D. L., and Bansal, R. R. 1975. "Linear Fresnel Lens Concentrators." Solar Energy 17:285. Nixon, G. 1977. "Cast Acrylic Fresnel Lens Solar Concentrator." Report of Swedlow, Inc., 12122 Western Ave., Garden Grove, CA. O'Neill, M. J. 1979. "A Unique New Fresnel Lens Solar Concentrator." International Solar Energy Congress, Atlanta, GA. Also U.S. patent No. 4,069,812. Solar Concentrator and Energy Collection System. Pitman, C. L. and Vant-Hull, L. L. 1982. "Atmospheric Transmission Model for a Solar Beam Propagating Between a Heliostat and a Receiver." Progr. Solar Energy p. 1267. Rabl, A. 1976. "Comparison of Solar Concentrators." Solar Energy 18:93. Riaz, M. 1976. "A Theory of Concentrators of Solar Energy on a Central Receiver for Electric Power Generation." J. Eng. Power 98:375. This reference derives closed analytical formulas for shading and blocking effects. Russell, J. L. 1974. "Central Station Solar Power." J. Power Eng. (November). Stromberg, R. P. 1975. "A Status Report on the Sandia Laboratories Solar Total Energy Program." Solar Energy 17:359. Tabor, H. 1958. "Stationary Mirror Systems for Solar Collectors." Solar Energy 2:27. Tabor H. 1966. "Mirror Boosters for Solar Collectors." Solar Energy 10:111. Tabor, H. and Zeimer, H. 1962. "Low Cost Focusing Collector for Solar Power Units." Solar Energy 6:55. Vant-Hull, L. L. 1976. "An Educated Ray Trace Approach to Solar Tower Optics." In Proceedings of the Society of Photo-Optical Instrumentation Engineers, Vol. 85, p. 111. Vant-Hull, L. L. and Hildebrandt, A. F. 1976. "Solar Thermal Power System Based on Optical Transmission." Solar Energy 18:31.
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Walton, J. D. 1980. "Development of the Spiral Fresnel Concentrator." Presented at International Symposium on Solar Thermal Power and Energy Systems, Marseille, France. Walzel, M. D., Lipps, F. W., and Vant-Hull, L. L. 1977. "A Solar Rux Density Calculation for a Solar Tower Concentrator Using a Two-Dimensional Hermite Function Expansion." Solar Energy 19:239. Winston, R. 1970. "Light Collection Within the Framework of Geometrical Optics." J. Opt. Soc. Am. 60:245. Winston, R. and Welford, W. T. 1979. "Geometrical Vector Flux and Some New Nonimaging Concentrators." /. Opt. Soc. Am. 69:532.
8. OPTICAL ANALYSIS AND OPTIMIZATION OF PARABOLIC REFLECTORS
Traditionally the optical analysis of solar concentrators has been carried out by means of computer ray trace programs [Biggs and Vittitoe, 1979; Schrenk, 1963; Treadwell, 1976]. Ray tracing is a microscopic method that can provide an enormous amount of detailed information but obscures functional relationships. Recognition of functional relationships is of utmost importance for the development of simplified design procedures. This chapter shows how all the relevant parameters for the optical design of parabolic solar concentrators can be obtained by a simple macroscopic approach. In most practical cases approximations are permissible whereby all quantities of interest can be determined from a few graphs. The method could be applied to all focusing collectors, but has so far been used only for parabolic reflectors. The method lends itself very well to the problem of optimizing the concentrator design, as shown by Rabl et al. [1982]. 8.1 ANALYTICAL FRAMEWORK In a real solar concentrator, rays are incident from a range of directions, covering the solar disk and possibly the circumsolar region, and are reflected by an imperfect reflector surface that causes further angular dispersion. From the point of view of the receiver, it does not matter whether the angular deviation of a ray from the design direction originates at the radiation source or at the reflector. This is illustrated in Fig. 8.1.1. In Fig. 8.1.la a ray from a point source S strikes the reflector at a point R and would reach point Q of the receiver if the reflector were perfect. A real reflector differs from the design slope by an error 0slope and thus the reflected ray reaches the receiver at Q', an angle 20s,ope away from Q. The same reflected ray would have resulted from a perfect reflector if the incident ray had come from S', an angle 2#slope away from the point source S, as shown in Fig. 8.Lib. In many cases the distribution of slope errors is nearly Gaussian, and the corresponding distributions are indicated by the dashed curves in Figs. 8.1.1 a and 8.1.Ib. This observation provides the basis for our approach. First, the optical errors are combined with the real source to yield what we call the effective source. The real source and the optical errors are characterized by angular distribution functions. Hence, the effective source J5efr (0) is also an angular 196
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197
Figure 8.1.1 Equivalence between (a) imperfect reflector with point source and (b) perfect reflector with smeared source.
distribution; it describes how much radiation is incident from the direction 6 on the aperture of a perfect reflector. As a brightness distribution 5eff (6) and 5Source (0) have units of W/m2 mrad in two dimensions (line-focus) and W/m2 sterad in three dimensions (point-focus). To calculate the radiation intercepted by the receiver, we define an angular acceptance function f (6) as that fraction of a uniform beam of parallel rays incident on the aperture at an angle 6 from the symmetry axis that would reach the receiver if the optics were perfect. /(0) accounts for off-axis aberrations of the parabola. With these definitions, the intensity of radiation on the aperture from the direction 6 and reaching the receiver is given by the product 5cn{0)/(0). The total amount of radiation 7in reaching the receiver is obtained by integration over all angles. For parabolic troughs one can work with two-dimensional distributions and 7in is
For parabolic dishes it is given by
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if one assumes azimuthal symmetry for the optical error distribution. (Otherwise, one would also have to integrate over the azimuthal direction.) In these equations, we have extended the limits of integration to infinity because in practice the angular distributions will be negligible beyond a few degrees from the symmetry axis. The angular acceptance function depends on receiver shape, concentration ratio, and rim angle; it is calculated in Section 8.2 for the most important configurations. The effective source distribution is calculated in Section 8.3, with particular attention to three-dimensional complications at large incidence angles which are not obvious from a cursory look at Fig. 8.1.1. The effective source is the mathematical convolution of the brightness distribution of the sun with all the optical error distributions. The final result for the intercepted radiation 7in is the convolution of effective source and angular acceptance function /(0), evaluated at zero incidence angle. Since the order of carrying out convolutions is arbitrary, one could equally well define a smeared angular acceptance function as convolution of/(0) and optical errors, and use the smeared angular acceptance function together with the real source in Eqs. (8.1.1) and (8.1.2).
8.2 ANGULAR ACCEPTANCE FUNCTION 8.2.1 Parabolic troughs For a parabolic trough of rim angle <j> with cylindrical receiver, the angular acceptance function can be calculated from the geometric relations in Fig. 8.2.1. The focal length is/ the aperture width is D, and the receiver tube diameter is d. The geometric concentration C is
The equation of the parabola is
Light rays incident at point P = (x, y) of the reflector hit the receiver, provided their angle of incidence 0 (measured from the optical axis) satisfies
6 = 0 V is the angle at which the rays reach the receiver tube tangentially. 6X decreases with x. Therefore, with x = D/2, Eq. (8.2.3) yields the largest
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199
Figure 8.2.1 Geometric relations for the calculation of the angular acceptance function. angle 0, for which all incident rays are accepted. This angle 0, (which in Chapter 7 has been called the acceptance half-angle of the parabolic trough) can be written in terms of rim angle and concentration as
For the angular acceptance function this implies that^T>cyl(0) = 1 for |0| < 0,. For incidence angles larger than 0, but smaller than 02, given by
only the central section of the aperture is effective, from — x, to xe, with
and the angular acceptance equals 2xg/D. For angles larger than 02 only direct hits are accepted, but this region will not be of interest here. It is convenient to express everything in terms of rim angle and concentration by means of the relation
Neglecting complications or inaccuracies that may arise for very low concentrations, very large incidence angles, or very small rim angles (cases which are not of interest for solar energy applications), the angular accep-
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Active Solar Collectors and Their Applications
tance function for the parabolic trough with cylindrical receiver, rim angle , and concentration C can be summarized as
with
C is assumed large enough (C > 3) to justify replacing sin 6 with 6 for \6\
<e2.
For the parabolic trough with a flat one-sided receiver, the angular acceptance function can be derived in a similar manner. The result is
This function is unaffected by the addition of a CPC second stage concentrator of acceptance half-angle <j> because such a CPC only prevents rays with 181 > 62 from reaching the receiver. For future reference we note that the angular acceptance function for parabolic troughs depends only on the product CO, not on C and 6 separately. The angular acceptance function is plotted schematically in Fig. 8.2.2. 8.2.2 Parabolic dishes For a parabolic dish with spherical receiver, the angular acceptance function is the square of Eq. (8.2.7), with the replacement of irCby 2\/C:
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201
Figure 8.2.2 Angular acceptance function for 4> = 90° trough with cylindrical receiver (schematic) (From Bendtetal. [1979]).
with
For a parabolic dish with flat one-sided receiver, the exact expression for the angular acceptance function is more complicated. For practical applications, however, the following polynomial fit is acceptable [Bendt and Rabl, 1981]:
The coefficients a,b,c,v\, and v2 depend on rim angle and are listed in Table 8.2.la; a more accurate four-term expansion with an additional term d(C62)3 is given in Table 8.2. Ib. Implicit in these equations and throughout this chapter is the assumption that the concentration is high enough (C > 10 for line-focus and C > 100 for point-focus concentrators) to permit the approximation of sin 0 by 0. For parabolic dishes the angular acceptance function depends only on the product (CO2), not on C and 0 separately.
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TABLE 8.2.1 Coefficients of Angular Acceptance Function for a Parabolic Dish with Flat Receiver, (a) Three-Term Expansion, (b) Four-Term Expansion.3 a.
b.
(deg.)
a
b
c
30 35 40 45 50 55 60
-1.8660 0.2309 0.8866 1.1738 1.2646 1.2466 1.2075
32.1042 8.7987 2.4950 0.0861 -0.7577 -0.9444 -0.9682
-89.5992 -23.7599 -8.0311 -2.7177 -0.8522 -0.2199 0.0309
(deg.)
a
b
c
d
30 31
-4.4647 -4.5916 -3.2651 -2.9694 -2.6752 -1.8817 -0.9984 -0.8803 -0.5546 -0.2971 -0.1744 0.0281 0.2034 0.3484 0.4679 0.5406 0.6616 0.7394 0.7824 0.8420 0.8969 0.9437 0.9788 1.0122 1.0388 1.0557 1.0762 1.0897 1.1025 1.1123 1.1166
65.5256 65.3298 48.6801 43.7084 39.2132 30.1432 20.3357 18.5920 15.3131 12.7096 11.2891 9.3233 7.7235 6.4039 5.3146 4.5794 3.6087 2.9432 2.5347 2.0125 1.5651 1.1781 0.8666 0.5920 0.3587 0.1827 0.0037 -0.1388 -0.2654 -0.3772 -0.4623
-231.6741 -230.3732 -165.7419 -144.6810 -126.3405 -94.2664 -60.3045 -54.0918 -44.0733 -36.1594 -31.6287 -25.9283 -21.4737 -17.8527 -14.9032 -12.8446 -10.4495 -8.7703 -7.7098 -6.4216 -5.3625 -4.4598 -3.7263 -3.1055 -2.5799 -2.1673 -1.7782 -1.4587 -1.1845 -0.9430 -0.7488
199.6752 215.6702 145.2926 125.8221 108.4094 76.2125 41.6597 37.6157 29.8485 23.7696 20.5410 16.2998 13.1345 10.6252 8.6317 7.2859 5.7546 4.7180 4.1138 3.3377 2.7276 2.2215 1.8195 1.4926 1.2208 1.0151 0.8258 0.6730 0.5464 0.4376 0.3516
4>
32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
"i 0.4355 0.4785 0.5053 0.5189 0.5179 0.4969 0.4645
V2
0.5342 0.6284 0.7226 0.8205 0.9231 1.0290 1.1407
"i 0.4365 0.4806 0.4536 0.4517 0.4598 0.4682 0.4708 0.4498 0.4757 0.4879 0.4758 0.4729 0.4948 0.4993 0.4999 0.4916 0.5010 0.5013 0.4972 0.4924 0.4924 0.4904 0.4852 0.4834 0.4793 0.4721 0.4686 0.4622 0.4575 0.4514 0.4425
V2
0.5346 0.5536 0.5726 0.5916 0.6107 0.6298 0.6473 0.6667 0.6862 0.7057 0.7255 0.7454 0.7653 0.7853 0.8055 0.8260 0.8463 0.8670 0.8892 0.9101 0.9313 0.9527 0.9743 0.9962 1.0183 1.0411 1.0637 1.0865 1.1096 1.1329 1.1565
"From Bendt and Rabl [1981].
8.3 EFFECTIVE SOURCE 8.3.1
Effective source for line-focus
The angular distribution of radiation from a real source like the sun is given by the functional dependence of brightness Bsoum(8in) on incidence angle.
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203
Solar brightness data are usually reported as radial distribution .Bradial(0) in W/m2 sterad, being measured from the center of the solar disk. For linefocus systems it is convenient to transform the radial distribution to a linear one according to
In the remainder of this subsection only the linear brightness function [in W/m 2 rad] is considered, and the subscripts for longitudinal and transverse are dropped. If the mirror slope errors are characterized by a normalized distribution function £(20slope) in units of rad"1, then the reflected intensity in the direction (measured from the design direction) is
where (6 — 0 in )/2 = 0slope is the slope error and Bsoarce(8-m) d6m is the intensity of radiation coming from an angular region of width d6m around 0in. Integrating over d&-m produces the equivalent effective source as convolution of E and B:
In a real collector there will be several statistically independent sources of optical errors: lack of perfect specularity, macroscopic surface deviations in position and slope, displacement of the receiver, and tracking errors. As discussed in Section 5.4, we shall assume that these errors can be described by Gaussian distributions, at least when averaged over time or over an entire collector or array of collectors. The distribution function E of optical errors in Eq. (8.3.3) should of course include the convolution of all the errors that are relevant in a given problem. Since the convolution of two Gaussians with zero mean and standard deviations a{ and a2 is again a Gaussian distribution, with standard deviation a2 = a\ + ffl, sun and mirror errors can be replaced by an effective source
The standard deviation ffoptical accounts for all optical errors and is given by the quadratic sum of the individual errors
In Fresnel reflectors, o-tracking must be multiplied by 2.
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Active Solar Collectors and Their Applications
In case the Gaussian model for the optical error is not acceptable, Eq. (8.3.4) must be replaced by the convolution of the appropriate error distribution functions. Certain reflector materials (for example, alzac) are characterized by the sum of two Gaussian distributions of relative weights Rt and R2 and widthsffspecuiarland o-spccl,,ar2. This type of material (which does not appear to be very practical for focusing solar collectors) can be dealt with by replacing the single Gaussian distribution in Eq. (8.3.4) with a sum two Gaussian distributions with weights R{ and R2 with widths aoptican and
Effective source for point-focus
For point-focus collectors the formulas are essentially the same as for the line-focus case, except that all integrations are two-dimensional, and brightness and error distributions have units of sterad~' instead of rad"1. Hence the effective source is given by
In most cases, the optical errors can be assumed to be approximately Gaussian, and the effective source becomes
We have indicated different standard deviations in different orthogonal directions x and y to allow for the possibility of error distributions without rotational symmetry. The orthogonal directions need not be x and y; in fact, a more realistic case is a distribution with different widths in azimuthal and radial directions. In any case, Eq. (8.3.5) for the addition of errors is to be used for each of the two orthogonal directions separately. Before closing this subsection, we point out that replacing contour errors by an effective source involves a slight approximation, and that Fig. 8.1.1 is misleading with regard to the situation in three dimensions. At oblique angles of incidence ^, a circular Gaussian distribution of contour errors with width a results in an elliptic Gaussian distribution of the reflected rays, with width a in the plane spanned by the reflected ray and by the surface normal and with width a cos \p in th eplane perpendicular to it, as shown in Section 5.4.3. The use of an effective source assumes that the distribution of the reflected rays remains a circular Gaussian for all points on the reflector. This
Optical Analysis and Optimization of Parabolic Reflectors
205
is exact at the apex of the parabola, where the incidence angle vanishes. At the rim of the parabola, the incidence angle reaches its maximum ^ = $/2, where $ is the rim angle. In most practical applications (certainly in solar energy), the rim angle will be less than 60°; hence the factor cos \j/ will be close to unity when averaged over the aperture, and the effective source is an excellent approximation. This is necessary in order to permit calculation of intercepted radiation by a formula as simple as Eq. (8.3.7). The approximation is conservative; i.e., it underpredicts the intercepted radiation. 8.4 FLUX AT RECEIVER AND INTERCEPT FACTOR The effective source function 5eff(0) gives the intensity of radiation (W/m2 rad) coming from the direction 6; it accounts correctly for the shape of the sun and for all optical errors. The angular acceptance function /(0) states how much of this radiation is transmitted to the receiver. The total flux intercepted by the receiver is obtained by multiplying these two functions and integrating over all incidence angles. For line focus this yields
The receiver size enters through the concentration ratio C in the angular acceptance function. Dividing Eq. (8.4.1) by the total incident flux
one obtains the intercept factor
which is useful because it is independent of the intensity. For point focus collectors the formulas are essentially the same, with the replacement of the linear integration by one over a radial variable:
The optical efficiency (relative to 7fc) is
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Active Solar Collectors and Their Applications
This formulation is equivalent to a detailed computer ray trace program. It is much simpler and faster, requiring at most a double integration. The relevant parameters and their interrelation are clearly identified. In many cases approximations can be made to the point where the explicit result can be presented in.graphical form. This is discussed in the following section. More accurate results for the intercept factor can be found in Section 8.6, along with an analysis of the effect of circumsolar radiation. Time-of-day effects for parabolic troughs can be treated exactly by evaluating Eq. (8.4.3) for each hour of the day using the projected sunshape discussed in Section 5.5. In most cases, however, a single calculation with an effective average sunshape for the whole day is sufficiently accurate. 8.5 APPROXIMATION OF SUNSHAPE BY GAUSSIAN DISTRIBUTION When the optical errors are large compared to the width of the sun, the intercept factor is insensitive to details of the sunshape. In that case it is convenient to approximate the sunshape by a Gaussian distribution. For line focus the formulas are
The resulting effective source is also a Gaussian distribution:
with width
The intercept factor 7 is denned as the ratio of the flux reaching the receiver over the incident beam irradiance Ib:
Since the angular acceptance is a function of CO only, yGatlK depends only on the product
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207
Figure 8.5.1 Intercept factor 7 versus
For the point-focus case the intercept factor for a Gaussian source is
where f(&) is the appropriate angular acceptance function, Eq. (8.2.11) or Eq. (8.2.14). For the case of a flat receiver it is plotted in Fig. 8.5.2. The accuracy of the Gaussian approximation has been tested by Bendt et al. [1979] and by Biggs and Vittitoe [1979] and found to be acceptable for most solar applications. The error in intercept factor is less than 1% if the optical error is larger than about 5 mrad for line focus, and 10 mrad for point-focus concentrators.
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Active Solar Collectors and Their Applications
Figure 8.5.2 Intercept factor 7 for parabolic dish with flat receiver geometric concentration C, and rim angle <j>, if source is Gaussian with width
Consider a parabolic dish with flat one-sided receiver and with the following parameters: A = C = $ = p = a =
10 m2 1000 45° 0.9 0.95 10 mrad
= aperture area = geometric concentration ratio = rim angle = reflectance of dish = absorptance of receiver = optical error
Calculate the power absorbed by the receiver when the beam irradiance is
and the rms width of the sun is
[this corresponds to average clear sky conditions according to Eq. (5.4.7)]. SOLUTION
The total beam width is
and the value of the abscissa in Fig. 8.5.2 is
Optical Analysis and Optimization of Parabolic Reflectors
209
With the $ = 45° curve one finds the corresponding intercept factor on the ordinate
Then the power absorbed by the receiver is
8.6 INTERCEPT FACTOR AND CIRCUMSOLAR RADIATION Due to atmospheric scattering, a significant amount of solar radiation arrives at the earth's surface from directions other than the sun. The radiation within a few degrees of the solar disk (the solar aureole) is called circumsolar radiation. Circumsolar radiation poses a problem for the analysis of concentrating collectors because, in general, the acceptance angle or field of view of a concentrator will differ markedly from that of the standard instruments available for measuring the direct or beam component Ib of solar radiation. By far the most common instrument that has been used and is likely to be used in the future for measuring Ih is the normal incidence pyrheliometer (NIP). Because it has a field of view characterized by an acceptance halfangle 10 times larger than the solar disk itself, it includes a sizable portion of circumsolar radiation. Determination of the distribution of circumsolar radiation is particularly important for collectors with high concentration because their field of view is considerably smaller than that of the pyrheliometer; hence predictions based on pyrheliometer data may overestimate the performance significantly, on the order of several percent. Since 1974 a systematic program to measure the distribution of circumsolar radiation has been carried out by the Lawrence Berkeley Laboratory [Grether et al, 1977, 1979]. Four circumsolar telescopes have been built that scan across the sun to measure the angular brightness distribution B(ff) with an angular resolution of 1.5 minutes of arc from the center of the sun to 0.5°, and 4.5 minutes of arc from 0.5-3.2°. They have been operated at several locations, including Albuquerque, N.M.; Argonne, IL.; Atlanta, GA.; China Lake, CA.; and Fort Hood, TX. Data for these locations are summarized in Table E. 1 of Appendix E. Basically, there are two questions concerning the effect of circumsolar radiation on concentrating solar collectors. The first is: Given the specific sunshape at a particular moment, what is the corresponding instantaneous performance of a concentrating collector? This question is important for verification of the detailed agreement between test data and optical analysis. It requires as input the solar brightness as measured by a circumsolar telescope during the test. The second question concerns the long term average decrease in performance due to the loss of circumsolar radiation. This is
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Active Solar Collectors and Their Applications
important for system performance prediction and economic analysis. In the following we describe a simple yet very accurate model for calculating the average effect of circumsolar radiation. For a derivation of this model and for details on calculating the instantaneous effect of circumsolar radiation the reader is referred to Rabl and Bendt [1982]. The simplest way to account for circumsolar effects on long term average performance is to start with the usual pyrheliometer data for the beam irradiance Ib and to calculate the intercepted irradiance:
with the intercept factor
Rav(X) is the long term average circumsolar ratio (in Table E.I) for the location and period in question; to a good approximation it is a linear function of the threshold X. For the locations where circumsolar data are available, Rm(X) for several thresholds and for each month has been calculated and listed in Table E. 1 . The circumsolar ratio for any other threshold value X can be found by linear interpolation. The denominator of Eq. (8.6.2) corrects for the fact that a pyrheliometer does not see all the circumsolar radiation out to angles of 3.2° that has been included in the calculation of Rm. The coefficients 7, av and 7f av needed in Eq. (8.6.2) are the intercept factors for radiation from the solar disk and from the circumsolar region, respectively, as calculated for the standard scan. This standard scan is the simple average over all circumsolar scans measured by LBL and it can be used for any location; it is plotted in Fig. 5.4.1 and listed in Table E.2 of Appendix E. The coefficients 7, av and (7, av — 7,., av) are calculated according to Eqs. (48) and (49) of Rabl and Bendt [1982]. The results are reproduced in Appendix E of this book, in Table E.3 for parabolic troughs, and in Table E.4 for parabolic dishes. A wide range of concentrator parameters is covered in these tables. Once these coefficients are known, an accurate calculation of the intercept factor in Eq. (8.6.2) is straightforward, as illustrated by the following example.
EXAMPLE 8.6.1
A parabolic dish with concentration C = 1000, rim angle 0 = 40°, and total optical error with rms width a = 1 mrad has the coefficients
Optical Analysis and Optimization of Parabolic Reflectors
211
and Suppose the operating threshold is X — 100 W/m2; the corresponding average circumsolar ratio is found by linear interpolation between the X = 50 W/m2 and the X = 150 W/m2 entries in Table E.I. One of the lowest ratios in the table occurred in Albuquerque, N.M., in May 1976, with .Rav(50 W/ m2) = 0.020 and #av(150 W/m2) = 0.019; hence, ^av(100 W/m2) = 0.0195. The corresponding intercept factor is, from Eq. (8.6.2),
with respect to the radiation /b measured by a pyrheliometer. The highest average circumsolar ratio observed so far is that for Argonne, IL., in August 1977. Interpolation yields ^av(100 W/m2) = 0.120, and the corresponding intercept factor is 7av = 0.931, some 4% lower than that for Albuquerque in
May 1976.
REFERENCES Bendt, P., Rabl, A., Gaul, H. W., Reed, K. A. 1979. "Optical Analysis and Optimization of Line Focus Solar Collectors." Report SERI/TR-36-092. Golden, CO: Solar Energy Research Institute. Bendt, P. and Rabl, A. 1981. "Optical Analysis of Point Focus Parabolic Radiation Concentrators." Appl. Opt. 20:674. Bendt, P., Gaul, H. W., and Rabl, A. 1980. "Determining the Optical Quality of Focussing Collectors Without Laser Ray Tracing." ASME J. Solar Energy Eng. 102:128. Biggs, F. and Vittitoe, C. N. 1979. "The Helios Model for the Optical Behavior of Reflecting Solar Concentrators." Report SAND 76-0347. Albuquerque, NM: Sandia Laboratories. Grether, D. F., Hunt, A., and Wahlig, M. 1977. "Techniques for Measuring Circumsolar Radiation." Report LBL-8345. Berkeley, CA: Lawrence Berkeley Laboratory. Grether, D. F., Evans, D., Hunt, A., and Wahlig, M. 1979. "Application of Circumsolar Measurements to Concentrating Collectors." In Proc. International Solar Energy Congress, Atlanta, GA. Pettit, R. B. 1977. "Characterization of the Reflected Beam Profile of Solar Mirror Materials." Solar Energy 19:733. Rabl, A. 1976. "Comparison of Solar Concentrators." Solar Energy 18:93. Rabl, A. and Bendt, P. 1982. "Effect of Circumsolar Radiation on Performance of Focussing Collectors." ASME J. Solar Energy Eng. 104:237. Rabl, A., Bendt, P., and Gaul, H. W. 1982. "Optimization of Parabolic Trough Collectors." Solar Energy 29:407. Schrenk, G. M. 1963. "Analysis of Solar Reflectors: Mathematical Theory and Methodology for Simulation of Real Reflectors." Report GMC-AO-EDR3693. Treadwell, G. W. 1976. "Design Considerations for Parabolic Cylindrical Solar Collectors." Report SAND 76-0082. Albuquerque, NM: Sandia Laboratories.
9. HEAT TRANSFER IN SOLAR COLLECTORS
Murphy's law applied to solar collectors: Heat transfer coefficients are small only if you don't want them to be.
9.1 GENERAL CONSIDERATIONS In most solar collectors all three heat transfer modes—conduction, convection, and radiation—are important, and an exact calculation of the heat loss can be very complicated. In this chapter, we develop not only the formulas needed for an accurate heat loss calculation, but we indicate shortcuts by which the computational effort can be minimized without much sacrifice in accuracy. To put the required accuracy into perspective, one should note that in many applications collector heat losses amount to a relatively small fraction of annual collected energy. For example, if the annual heat loss is 30% of the annual collected energy, then a 10% error in estimating the heat loss causes only a 3% error in collected energy. In addition to required accuracy, one has to keep in mind achievable accuracy when the input is uncertain. Frequently, one does not know the emissivities very well; this is particularly true of selective coatings where emissivities may have relative uncertainties on the order of 10%-50% depending on coating type and manufacture. Convective heat transfer is difficult to calculate with better than 20% accuracy, even under the best of circumstances. Some collectors involve geometric configurations for which convective heat transfer coefficients have never been measured or calculated. Even though the achievable overall accuracy of a heat loss calculation may be quite low, there are situations where one would like to model certain details with much greater precision. For example, when comparing the effectiveness of different selective coatings one needs to evaluate changes in heat loss due to changes in emissivity. Such changes can be significant even if they are much smaller than the overall accuracy of the calculation. Accordingly we present in this chapter not only approximations that will be adequate for most applications but also more complicated and accurate formulas that are useful for detailed sensitivity studies. The problems that are involved in a heat loss calculation are illustrated in Fig. 9.1.1 for a flat plate collector with two nonabsorbing covers. The 212
Heat Transfer in Solar Collectors
213
Figure 9.1.1 Thermal network for flat plate collector with two covers that absorb no solar radiation but are opaque to infrared, (a) Detailed network in terms of resistances for convenction, radiation, and conduction, (b) Equivalent simplified network in terms of t/fronl and C/back. (c) Equivalent network in terms of collector U value U.
thermal network shown in this figure already assumes some approximations because it is one dimensional and neglects edge effects such as conduction through the frame from absorber to glazing. Temperature variations across the absorber plate are also neglected for the moment; they will be analyzed in Chapter 10. In a one-dimensional model each surface (absorber, first glazing, second glazing, collector back) is treated as one node in the thermal network. Heat losses through the top involve convection and radiation
214
Active Solar Collectors and Their Applications
between each node. This is indicated by parallel resistances for convection and for radiation. Ambient is represented by two nodes, air temperature ra,r for convection and the sky at rsky for radiation, because these two temperatures can be quite different. For the loss through the back, one node at rair is assumed; this could be different if the collector is mounted flush on the roof of a house. Since the equations for convective and for radiative heat transfer are nonlinear, an exact calculation requires iterations to determine the temperature of each node. In view of the accuracies that are required or achievable, such a computational effort is rarely justified. Instead, one can linearize the thermal network by approximating all resistances by temperature independent constants as in Fig. 9.1.1b. To demonstrate the linearization of the heat transfer equations, consider two parallel surfaces at temperatures T, and T2. The radiation heat flow per unit area is given by
where e is the effective emissivity [see Eq. (9.3.2)] and
is the Stefan-Boltzmann constant. This can be written in the form
with a radiative conductance /2rad denned by
where
is the mean temperature. Usually one will obtain a good approximation for 4ad if one makes a reasonable guess for the mean temperature and then proceeds with a constant /zrad. For the convective heat flow qcmv one can define a convective heat transfer coefficient hmm such that
h,onv is a function of temperatures (T, and T2), geometry, and fluid properties. Often the variation of /2L,OI,V with temperature is not very pronounced,
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Heat Transfer in Solar Collectors
and one can approximate it by a constant. Formulas for /zconv will be presented in later sections of this chapter. Ranges of typical values are indicated in Table 9.1.1. Usually one represents correlations for convection in terms of a dimensionless number, the Nusselt number Nu, denned such as to make
where k is the conductivity of the fluid and L is the characteristic length. Convection is notoriously difficult to calculate reliably, especially in configuration that are more complex than plates and tubes. Since the resistance between two nodes is really a function of the temperature of the nodes, one must start with an initial guess for the temperatures of all intermediate nodes. Fortunately, even a sizeable error in the initial guess will have little effect on the end result. This will be demonstrated with some examples in Section 9.9.3. Once the resistances have been approximated by constants, it is straightforward to calculate the overall heat loss conductance of the collector. This is shown in Fig. 9.1.1 b, where the network for the collector top is lumped together into a single conductance UfmM and similarly the network for the collector back is lumped into a conductance C/back. If, as will frequently by the case, the ambient temperatures in front and in back of the collector are more or less the same, one can also lump these two conductances together into a single conductance, the collector U value, as shown in Fig. 9.1.1c. After carrying out such a calculation, it is, of course, advisable to check whether the initial guess for the intermediate temperatures was indeed good enough not to cause undue errors. It is impossible, in the brief space of this chapter to analyze the heat transfer in all collector types that have been built. However, most collectors can be approximated in terms of certain standard geometric configurations, for which the heat transfer equations are reasonably well known. The most important standard configurations are the following: 1. Heat transfer to ambient: Typically this involves the heat flow from flat or tubular surfaces by radiation to the sky and by natural or forced convection (wind) to an infinite surrounding volume of air. TABLE 9.1.1 Approximate Ranges of Convective Heat Transfer Coefficients8 W/m 2 K Air, free convection Superheated steam or air, forced convection Oil, forced convection Water, forced convection Water, boiling Steam, condensing "From Kreith and Kreider [1980].
6-30 30-300 60-1800 300-6000 3000-60,000 6000-120,000
216
Active Solar Collectors and Their Applications
Figure 9.1.2 Planar receiver configuration, (a) Receiver relative to reflector, (b) Receiver.
2. Planar configuration: Heat transfer between flat parallel surfaces; e.g., between a flat absorber and a cover glass of approximately the same size. This is relevant not only for flat plate collectors, but also for some focusing collectors as shown in Fig. 9.1.2. Figure 9.1.2b shows receiver; Fig. 9.1.2a shows typical orientation relative to focusing parabolic dish or trough reflector. 3. Tubular configuration: Tubular absorber, surrounded by a glass tube, as shown in Fig. 9.1.3b and Fig. 9.1.3a, shows typical orientation relative to focusing parabolic trough reflector. 4. Concentrating configuration: Flat or tubular absorber, surrounded by insulated sides and a flat or curved cover that is larger than the absorber, as indicated schematically in Fig. 9.1.4. Practical realizations include the V-trough or CPC with flat absorber, the CPC with tubular absorber, Fig. 9.1.5, and the Fresnel lens with flat absorber. 5. Heat transfer inside ducts and pipes. 6. Conduction through insulation.
Figure 9.1.3 Tubular receiver configuration with concentric glass cover, (a) Receiver relative to reflector, (b) Receiver.
Heat Transfer in Solar Collectors
217
Figure 9.1.4 Concentrating receiver configuration, schematic. Cover may be flat glass or Fresnel lens; side walls may be straight or curved.
In the following sections we give formulas that can be used for calculating the heat transfer for each of these basic configurations. In most collectors, more than one configuration is relevant. For example, the heat loss from the nonevacuated CPC in Fig. 9.1.5 is found by calculating the thermal resistance Rlube between the absorber and the glass tube based on configuration 3, then adding it to the resistance RmK between the glass tube and the cover, based on configuration 4, and finally adding the resistance from cover to ambient (configuration 1). Configurations 2 and 3 have been thoroughly covered in the heat transfer literature, and fairly reliable calculations can be made. Configuration 4, on the other hand, has not been well explored, and the lack of sufficient data on natural convection may introduce errors up to 50% in the calculation of the heat loss. Two further configurations, honeycombs and transpired absorbers, are discussed in Section 9.8. 9.2
HEAT FLOW TO AMBIENT
Heat transfer from collector to ambient occurs by radiation and by convection. If air, sky, and ground have the same temperature ramb, then the radiative heat flow from cover to ambient is simply
Figure 9.1.5 CPC reflector coupled to tubular receiver.
218
Active Solar Collectors and Their Applications
This assumption is usually made in practice. As for emissivities, one takes ambient to be a blackbody while a typical glass cover has an infrared emissivity ecov = 0.88 [Tabor, 1958]. Without these assumptions the calculation becomes more complicated. For example, consider a flat plate collector at tilt /3 if Tsky ^ T^^ (but still assuming sky and ground to be blackbodies). Then Eq. (9.2.1) is replaced by
The sky temperature is the equivalent blackbody temperature of the atmosphere for radiation transfer from the earth's surface to the atmosphere. Under humid or overcast conditions, TAy can be close to the air temperature, but when the humidity is low and the sky clear, the sky temperature can be as much as 30°C below air temperature. Several approximate formulas for sky temperature can be found in the literature. Duffie and Beckman [1980] quote a formula from Bliss that depends on both air temperature rair and dew point temperature Tdp (all in degrees Kelvin! [°K]):
The effect of 7"sky on a collector is reduced if the collector is tilted and sees the ground; clouds mask the sky and reduce the effect further. The difference between Tmr and rsky is on the order of 10°C in hot humid weather and on the order of 30°C in cold dry weather. Even though the difference between T"sky and Tmr is undeniable, in practice few people pay any attention to it when analyzing collector performance. First of all, variation of 7"sky has an appreciable effect on collector efficiency (on the order of a few percent) only for collectors with high heat loss coefficient (greater than approximately 5 W/m 2 °C). Second, it would be quite costly to measure the sensitivity of a collector to sky temperature: one would have to repeat the collector tests on days with different Tsky but everything else being the same (otherwise the effect of rsky would be masked by variations in air temperature, insolation, and wind speed). Waiting for days with the right conditions and sorting out rsky effects from other second order corrections would be very time consuming, and it is not done under standard collector test procedures. Third, actual data for Tsky are rare. In view of this, one assumes that the relation between 7"sky and Tmr during collector testing is typical of actual operating conditions. Collector test data are reported as a function of ambient temperature Tamb, which is taken as air temperature. Therefore, when a system analysis is based on test data and ambient air temperatures, the effect of sky temperature is implicitly accounted for in an average sense. For high-temperature collectors, the effect
Heat Transfer in Solar Collectors
219
of sky temperature is so small that one can safely set 7*sky = T& even when calculating the instantaneous efficiency. (For example, if rabs = 800 K then a change from Tsky = 300 to 270 K would increase the radiative heat loss by only 1 % if the absorber faced the sky. Actually the sensitivity to rsky is smaller because part of the absorber faces mirrors or ground.) For the convective heat transfer from solar collectors many and often conflicting correlations have been published in the literature. Even under the most controlled conditions convective heat transfer has large uncertainties, typically 10% or more. In real life the convection from a solar collector is rarely as simple as the idealized situations treated in laboratory experiments or in theoretical calculations. A real collector is surrounded by all kinds of objects, trees, buildings, other collectors, and so forth, that interfere with the free flow of air. An accurate experimental determination of convective heat transfer from solar collectors appears to be a hopeless task in view of all the possible variables and configurations that would have to be considered. Fortunately even a sizeable error in this convective heat transfer coefficient will usually not have an overriding effect on the calculation of annual collectible energy. As an example, let us consider a single-cover selective flat plate collector for which the front loss coefficient £/from is given in Fig. 9.9.4d-9.9.4f. For an absorber temperature rabs = 40°C and air temperature ramb = 10°C the loss coefficient is U(rom = 2.95 W/m2 K when the convective heat transfer coefficient from cover to air is /zwind = 1 0 W/m2 K. If //wimi were twice as large, /zwind = 20 W/m 2 K, f/front would increase to 3.2 W/m2 K, while it would decrease to 2.6 W/m 2 K if /zwind decreases to 5 W/m2 K. For this collector a ±50% variation in /zwind causes only a ±10% variation in C/from which, depending on operating conditions, may change annual collected energy by perhaps 2%-5%. The sensitivity of a collector to /zwind increases with C/front. At one extreme is the unglazed collector used for swimming pool heating; for this collector the front loss coefficient
is obviously very sensitive to /2wind since the radiative term contributes only about 5 W/m2 K. At the other extreme are evacuated collectors for which uncertainties in /zwind have negligible effect. For parabolic troughs with nonevacuated single-glazed selective absorber (the most common design), the sensitivity to hmnd is about the same as for the single-glazed selective flat plate discussed above. Many high-temperature point-focus collectors use unglazed nonselective absorbers and are hence more sensitive to /zwind; however; convection is relatively less important than radiation at high temperature. In light of this we make the following recommendations, escalating in complexity according to desired accuracy and detail. At the simplest level one can assume a single effective heat transfer coefficient A amb for both convection and radiation. Setting 7"sky = Tmr = 7"amb one then gets a heat flow
220
Active Solar Collectors and Their Applications
from cover to ambient. The ASHRAE Handbook of Fundamentals [ASHRAE, 1977] recommends approximately the following values:
A year-round average value of
appears to be a reasonable choice. This approximation will usually be acceptable for evacuated collectors and for collectors with selective coatings. In case of doubt one should repeat the calculation with a different number for /zamb to see whether this is worth worrying about. If greater accuracy is desired one has to calculate convection and radiation separately. Let us designate the convection coefficient from cover to ambient by /zwind (even when there is only natural convection). For natural convection the Nusselt number [see Eq. (9.1.5)] for the heat transfer coefficient is given by a correlation of the form
where a and b are the coefficients listed in Table 9.2.1. Gr and Pr are the Grashoffand Prandtl numbers, respectively; they are given by
TABLE 9.2.1 Coefficients of Nusselt Number Relation Nu = b (Ra)a for Natural Convection from Flat Surface3 Range of Ra
b
a
Vertical plate Laminar flow Turbulent flow
104-109 109-1012
0.59 0.13
y, %
Horizontal plate Laminar flow Turbulent flow
105-2 X 107 2 X 10 7 -3X 10'°
0.54 0.14
Yt y.
"From McAdams [1954].
Heat Transfer in Solar Collectors
221
with k = thermal conductivity [W/m °K],
L - characteristic length (for flat plate taken as length of plate) [m], g = earth's acceleration (= 9.80 m/sec2), j3T = volumetric expansion coefficient (for air one can assume /3T = I/T with T in degrees Kelvin), v = kinematic viscosity [m2/sec], a = thermal diffusivity [m2/sec]. For forced convection the Nusselt number depends on wind speed v via the Reynolds number Re
To a lesser extent it also depends on the angle of attack of the wind, on the shape of the plate, and on surrounding structures. Some of these variables have been investigated in a series of elegant experiments by Sparrow and his co-workers [Sparrow and Tien, 1977; Sparrow, Ramsey, and Mass, 1979; Sparrow, Nelson, and Tao, 1982]. These measurements covered the laminar flow regime from Re = 20,000-100,000. If one is willing to disregard variations (up to about ± 10%) due to shape and orientation of the plate, then Sparrow et al. [1979] recommend as a good compromise the correlation
with the characteristic length L denned as 4 X area divided by circumference. More detailed information can be found in these references. For high wind speeds or large plate sizes the flow is turbulent, and can use the relation [Kreith, 1973]
In case of low wind speeds it is advisable to calculate both natural and forced convection and take the larger of these two numbers. For forced convection from tubes Duffie and Beckman [1980] recommend that one use the relation of McAdams [1954]:
222
Active Solar Collectors and Their Applications
They further recommend that these values be increased by 25% to account for outdoor conditions. 9.3 PLANAR CONFIGURATION Radiative heat transfer between two infinite parallel plates 1 and 2 per unit area is given by
with
Numerical values are listed in Table 9.3.1. T and t designate the temperature and emissivities, with appropriate subscripts, of each surface. For glass Tabor [1958] recommends an emissivity %iass = 0.88. In flat plate collectors one can usually assume Eq. (9.3.2) without worrying about edge effects. In some concentrating collectors the plate width may not be large compared to the plate separation, and the effective emissivity may have to be modified to account for a limited view factor [Sparrow and Cess, 1978]. With regard to convection between parallel plates, many correlations can be found in the literature. Tabor [1958] conducted a survey and concluded that for flat plate solar collectors the most reliable correlations as of 1958 were contained in Report 32 of the U.S. Home Finance Agency [1954]. According to Tabor the convective heat transfer coefficient between two parallel plates separated by L
is given by the following expression for the Nusselt number:
TABL E 9.3.1 Ef Fective Emissivity «eff ==
<W(«i +
C2 -
«,£2)
(2
0.05
0.1
0.15
0.2
0.3
0.5
0.9
0.95
0.1 0.2 0.88 0.95
0.0345 0.0417 0.0497 0.0499
0.0526 0.0714 0.0987 0.0995
0.0638 0.0938 0.1470 0.1488
0.0714 0.1111 0.1947 0.1979
0.0811 0.1364 0.2882 0.2953
0.0909 0.1667 0.4681 0.4872
0.0989 0.1957 0.8016 0.8593
0.0995 0.1979 0.8410 0.9048
Heat Transfer in Solar Collectors
223
where a and b are the constants given in Table 9.3.2, and Ra is the Rayleigh number, given by Eqs. (9.2.7) through (9.2.9), with characteristic length L. In convective heat transfer most of the thermal resistance is due to conduction across a more or less stagnant fluid film at the surface. In natural convection between two parallel plates there are two such films, one at each surface. If the plates are sufficiently close together, the two films may overlap and convection is inhibited. How much convection occurs depends on the values of Ra in Eq. (9.3.4). When Ra < 1700 there is only conduction and the Nusselt number equals unity. In solar collector design one typically wants to minimize the heat transfer between the two plates. Thus it is important to consider the variation of the convective heat transfer coefficient /Zi_ 2 , C onv with plate separation L. This is shown in Fig. 9.3.1. In this figure the coordinates for h and L contain temperature correction factors b $2, and <j!>3 given by Eqs. (9.3.5) through (9.3.7), with Tm = (Tt + T2)/2 in °K,
(Figure 9.3.1 is actually based on more recent correlations of Hollands et al. [1976] [See Eq. (9.3.8)] below, and includes a dependence on collector tilt.) For small gaps L there is only conduction and h behaves like k/L. As L is increased, convection sets in, typically at critical values around L = 0.7-1 cm. The heat transfer coefficient has a local minimum just as convection begins, and grows again as L increases beyond the critical value. A local maximum 10%-20% above the local minimum is reached when L is about 30%-50% larger than the critical value. Finally L decreases again towards an asymptotic limit somewhat lower than the local minimum at the critical value. The lessons for collector design are clear. If there is no cost penalty for large spacings, one should choose L as large as possible to minimize h. TABLE 9.3.2 Nusselt Number for Natural Convection Between Parallel Plates" Horizontal planes, heat flow upwards Nu = 0.168(GrPr)°-281 = 0.152 Or0-281 for Gr from 10M07 45° planes, heat flow upwards Nu = 0.102(GrPr)a3l° = 0.0925 Gr0-310 for Gr from 104-107 Vertical planes Nu = 0.0685(GrPr) 0327 = 0.0616 Gr0327 for Gr from 1.5 X 105-107 Nu = 0.0369(GrPr)0381 = 0.0326 Gr0-381 for Gr from 1.5 X 104-1.5 X 105 "The second set of numbers is for air at 10°C and atmospheric pressure (From Tabor [1958]).
224
Active Solar Collectors and Their Applications
Figure 9.3.1 Plot of the free convection heat transfer coefficient across an air gap versus gap spacing L. The extrapolated values are all right for /? = 0 but for 0 = 70° they underpredict by about 10% (Hollands, personal communication, May 1983) (From Hollands et al. [1976]).
However, in practice, economic constraints may place a premium on compactness, and hence one may prefer to hit the critical separation just at the onset of convection where h has its local minimum. With variable operating conditions this is not always possible because the position of the minimum depends on operating temperatures through the correction factors 02 and 03-
If the effects of collector tilt are to be included, a recent correlation by Hollands et al. [1976] can be used in lieu of Eq. (9.3.4):
where the + subscript indicates that only positive values of the terms in the square brackets are to be used (i.e., use zero if the term is negative). It is good for a tilt angle range from /? = 0° to ft = 75°, which covers most solar applications of interest. This correlation is plotted in Fig. 9.3.2; for easier use a second scale for air only is included with correction factors Fl and F2 in Fig. 9.3.3. Since collectors have a finite extent, the ratio of plate separation over plate width, also called aspect ratio, may have an effect on the heat transfer. This is shown in Fig. 9.3.4, again with a second scale for air with the correction factors FI and F2 of Fig. 9.3.3. The use of these graphs is illustrated in Example 9.9.2 in Section 9.9.
Heat Transfer in Solar Collectors
225
Figure 9.3.2 Nusselt number as a function of Rayleigh number for free convection heat transfer between parallel flat plates at various slopes (From Duffle and Beckman [1980]; reprinted by permission of John Wiley & Sons). 9.4 TUBULAR CONFIGURATION
The radiative heat transfer between two concentric tubes is also given by Eq. (9.3.1) with eeff of Eq. (9.3.2) if the outer tube is a specular reflector of thermal radiation. This is the typical case where a glass tube encloses an absorber tube.
Figure 9.3.3 Air property corrections F, and F2 for use with Fig. 9.3.2 (From Tabor [1958], as adapted by Duffie and Beckman [1980]; reprinted by permission of John Wiley & Sons).
226
Active Solar Collectors and Their Applications
Figure 9.3.4 Nusselt number as a function of aspect ratio A and of Rayleigh number for free convection heat transfer between vertical flat plates (From Duffie and Beckman [1980]; reprinted by permission of John Wiley & Sons).
The natural convection between concentric tubes can be calculated from a correlation by Kuehn and Goldstein [1978]. If the inner and outer diameters of the annulus are D, and Dm then the convective heat flow per unit surface area of the absorber or inner tube is
with the convective heat transfer coefficient
The Nusselt number is obtained from the following equations:
Heat Transfer in Solar Collectors
227
Qualitatively the variation of the heat transfer coefficient with annulus width is similar to Fig. 9.3.1. With tubular absorbers it is usually desirable to make the annulus as small as possible without of course causing excessive conduction heat transfer. Hence one will try to hit the local minimum; this is shown specifically for tubes in Fig. 9.9.1. In practice this corresponds to an annulus of width 0.7-1 cm for receivers of parabolic trough collectors.
9.5 CONCENTRATOR CONFIGURATION Heat transfer coefficients for the concentrator configuration are difficult to determine. The problem is complicated by the interaction between all three heat transfer modes: conduction (through the reflector side wall), convection, and radiation. In evacuated collectors fairly accurate results can be obtained quite easily. Convection, however, is very difficult to calculate for concentrator configurations. Recently convection correlations for CPCs with flat one-sided absorbers have been published by Abdel-Khalik et al. [1978], but their applicability may be limited since the interaction with the sidewalls depends on conductive and radiative effects which in turn depend on design details, materials, and construction. In the following we therefore present only some approximate formulas; for the analysis of nonevacuated CPC collectors [Rabl, 1976, 1980] they have predicted the convective heat loss coefficient with acceptable accuracy (within about 20%). The radiative heat flow from absorber to cover in the concentrator configuration is given by
where the effective emittance €eff of the absorber depends on geometry and on the emittances of absorber, side wall, and cover. Rabl [1976] has investigated this functional dependence of eefl in full detail for the CPC. As it turns out, there is no need to worry about much detail. In practice the cover1 will have tcov w 0.9 and the side wall (or reflector) will have either a metallic surface (aluminum) with low infrared absorptance, ewa]1 < 0.1, or else a nonmetallic surface with high infrared absorptance, ewaM > 0.9. In these cases ecff is almost independent of ewa,, and of CPC shape, and it is well approxi'In this configuration «cov is slightly larger than the value 0.88 recommended by Tabor for the flat-plate case, because the angles of incidence are restricted.
228
Active Solar Collectors and Their Applications
mated by Eq. (9.3.2):
Since reflector shape and ewa], do not matter for a wide range of CPC configurations, Eq. (9.5.2) appears to be a good approximation for all concentrator configurations with ewan below 0.1 or above 0.9. To estimate convection Rabl [1976] used the correlations for natural convection from flat surfaces in Table 9.2.1. These correlations were used separately at the absorber and at the cover, calculating the heat flow from absorber to the air and then from the air to the cover. The properties of air were parametrized by the following expressions, which provide a good fit over the range from 250 K to 600 K: for the expansion coefficient
for the kinematic viscosity
for the conductivity
The resulting formula for the convective heat flow (in the laminar regime) from absorber to cover is
where
and
is the geometric concentration ratio; d^ and dcov are the widths of absorber and cover. The constant b is found in Table 9.2.1. With b = 0.56 (for imme-
Heat Transfer in Solar Collectors
229
diate collector tilt), the grouping of constants in Eq. (9.5.6) has the value
Analysis of heat transfer in the concentrator configuration is further complicated by conduction of heat from the absorber into the side walls, and by absorption of solar radiation by the side walls. The latter mechanism raises the wall temperature and thus lowers the heat loss from absorber plate to cover. This can be analyzed most conveniently by the method of Section 9.10. Conduction of heat from the absorber into the reflector wall, on the other hand, can significantly increase the heat loss. This can pose a serious problem, since aluminum, the preferred reflector material by virtue of its durability, also has one of the highest thermal conductivities. If sheet aluminum is to be used in this configuration, it should therefore be as thin as possible, and a slight air gap between the absorber and the reflector is recommended.
9.6 HEAT TRANSFER INSIDE TUBES AND DUCTS
In most solar collectors the heat is extracted by means of a heat transfer fluid flowing through tubes or ducts. Hence one needs to analyze the heat flow from tube wall to fluid. Usually the conductance of the wall material is large enough to permit the approximation that the wall is isothermal. The heat flow from wall to fluid occurs by convection and is described by the convective coefficient h:
where k is the conductivity of the fluid and Dh the hydraulic diameter defined by
The Nusselt number Nu depends on whether the flow is laminar or turbulent and whether the tube is short or long. Kreith and Kreider [1980] recommend the correlations for Nu listed below, with the following notation: L = length of tube or duct M = dynamic viscosity of fluid (with subscripts b for viscosity at bulk, w for viscosity at wall temperature) v = /i/p = kinematic viscosity (p = density) u = velocity
230
Active Solar Collectors and Their Applications
Re/, = vD/v = Reynolds number Pr = v/a = Prandtl number a = thermal diffusivity Unless otherwise indicated by the subscripts b and w, the fluid properties are to be evaluated at the arithmetic mean of the fluid temperatures at inlet wall, inlet bulk, outlet wall, and outlet bulk. For laminar flow the Nusselt number is
in long tubes (Re Pr Dh/L > 10), while in short tubes (Re Pr Dh/L > 100) it is given by
For turbulent flow for long tubes (L/Dh > 60) and Pr between 0.7-700 the Nusselt number is
For turbulent flow in short tubes (L/Dh < 60) the last equation is to be multiplied by the correction factor [1 + Dh/L]°-7. Since the above correlations assume uniform wall temperature, they are not applicable to collectors where the fluid flows between two parallel plates: the heated absorber plate and an insulated backplate. This is a typical arrangement for flat plate air collectors, and the following formula has been recommended for this case [Kreith and Kreider, 1980]:
Note that Dh is twice the plate separation. In many applications the heat transfer coefficient h need not be calculated with great accuracy and the following expressions can be used for air or water at atmospheric pressure and in the temperature range of 300-380 K [Kreith and Kreider, 1980]. These expressions are dimensional and proper units must be used, in particular, D is in m, v in m/sec, Tin K, and h in W/ nr K. For turbulent air in long ducts one can use
Heat Transfer in Solar Collectors
231
For turbulent air flow between two parallel plates separated by a distance D, one plate heated, the equation is
Finally for turbulent flow of water through a long duct of inside diameter D the heat transfer coefficient is
9.7 CONDUCTION Conductive heat losses can be calculated easily for certain idealized situations. For example, the heat flow across an infinite uniform slab of thickness d, area A, and conductivity k is
where AT is the temperature difference across the slab. Another important case is the heat flow across the wall and/or insulation of a pipe. It is given by
where L is the length of the pipe, and r0 and rt are the inner and outer radii of the material. If there are several layers of different materials, these equations have to be used for each layer, in series. The resistance from the outer surface to ambient also has to be added in series.2 When calculating conductive heat losses of solar collectors, one should keep in mind that thermal conductivities increase with temperature, as shown in Fig. 13.1.3. Also the conductivities of insulating materials are usually quoted for dry conditions. For wet insulation the heat flow can be much larger, in particular, if the heatpipe effect can occur. In some collectors the geometry is not simple enough to permit a closed form solution of conductive heat losses. This is likely to be the case for receivers of concentrating collectors, because the receiver is relatively small compared to the thickness of a reasonable amount of insulation. In such cases one may have to resort to numerical solutions. Actually this problem arises even in flat plate collectors. As Tabor [1958] has emphasized, for typ2 When adding insulation to a pipe, one increases the outer surface area from which heat is lost to ambient. Depending on the parameters involved the heat loss may actually increase when insulation is added (e.g., see Kreith [1973]).
232
Active Solar Collectors and Their Applications
ical collectors the losses through the sides are not negligible. Tabor also has developed a simple approximation for taking this effect into account. First the heat loss through the front of the collector is treated as if it occurred by conduction through the same material as the insulation in the back and sides of the collector. An equivalent insulation thickness d(TOM is denned such that the frontal heat loss is
where L, and L2 are the length and width of the absorber plate, k is the conductivity, and AT the temperature difference across the insulation. In terms of the frontal collector U value d{TOM is given by
Thus the absorber plate is considered to be immersed in a rectangular box full of insulating material as shown in Fig. 9.7.1. The insulation thickness in the back is dback, and the width of the insulation at the edges is d^. For the configuration in Fig. 9.7.1 the total heat flow can be written in the form
The square bracket is the correction for finite plate size; it is stated in terms of the "characteristic" width
and an edge coefficient S.
Figure 9.7.1 Dimensions dfroM, dback, dsidc, and L = L, or L2 for analysis of conductive heat losses in flat plate collector.
233
Heat Transfer in Solar Collectors
The back loss ()back is the difference between Eqs. (9.7.5) and (9.7.3):
The U value (7back for the conductive heat loss is denned such that
Introducing a modified edge correction factor S' as
and readily obtains from Eq. (9.7.7) the result
The square bracket is the edge loss correction factor relative to a collector of infinite width. The coefficients S and S' depend on the ratios d^/d^ and dfromMack, and numerical results are listed in Table 9.7.1. These numbers have been calculated by H. Weinberger, as reported by Tabor [1958]. Several conclusions can be drawn from Table 9.7.1. Firstly, the edge coefficients vary strongly with dside/dback if this ratio is less than unity, but when rfside is increased beyond rfback, the effect on S and S' becomes small. This suggests that the optimal choice for the side insulation will be in the vicinity of
TABLE 9.7.1 Correction Factors S and S' for Conductive Heat Loss of Flat Plate Collector3 "front/"back
"side/"back ~~ °°
2
IX
1
1.24 1.86
1.28 1.92
1.43 2.15
/«
M
1.92 2.88
2.67 4.0
3.52 5.3
/2
2
S = 1.22 5' = 1.83
1
5 = 0.882 0.882 5' = 1.76 1.77
0.893 1.79
0.938 1.88
1.179 2.36
1.65 3.3
2.35 4.7
S = 0.611 0.615 5' = 1.83 1.84
0.616 1.85
0.62 1.86
0.71 2.13
0.96 2.88
1.33 4.0
1 2
"From Tabor [1958].
234
Active Solar Collectors and Their Applications
Secondly, in the region 6?sidc/'<4ack w 1 the correction factor 5" is quite insensitive to the value of dfmnt/dback. Thirdly, the edge correction is not negligible for typical collectors, as shown by the following example. EXAMPLE 9.7.1 A flat plate collector has abosrber area 1 m X 2 m and front loss coefficient tyrant = 5 W/m2 K. It is to be insulated with fiberglass of conductivity k = 0.05 W/m K. The insulation thickness at back and sides is rfback = <4de = 8 cm. Find the conductive U value for the heat losses through back and sides. SOLUTION
The characteristic length is
The value of dfront is from Eq. (9.7.4)
From Table 9.7.1 we estimate the edge correction factor to be S' = 1.86 (even though d(mm/dback = % is much smaller than %, S' does not change much in this region, and 1.86 is a conservative value). This yields the result
This value is 22% higher than for an infinite plate. The total U value for this collector is
9.8 OTHER CONFIGURATIONS In this section we describe two absorber designs that can be very effective in reducing heat losses: honeycombs and transpired absorbers. A honeycomb between absorber and cover can suppress convection if it has the right dimensions. The honeycomb material can be either reflective or transparent but it should be nonabsorbing for solar radiation.3 Absorptivity for infrared radiation may or may not be desirable, depending on the 3 Absorption of solar radiation tends to increase with incidence angle, hence it is advisable to check the incidence angle modifier of a honeycomb collector before using it.
Heat Transfer in Solar Collectors
235
details of radiative, convective, and conductive heat transfer in the honeycomb. Of course the thickness and/or conductivity of the material should be low enough to avoid excessive conduction losses. The heat transfer in honeycombs has been studied by Hollands [1965], Buchberg, Catton, and Edwards [1976], Felland and Edwards [1978], Meyer et al. [1978], and others. Honeycombs can be used both in flat plates [Buchberg and Edwards, 1976] and in the receivers of concentrating collectors [Francia, 1961]. Perhaps the simplest kind of honeycomb, both in terms of analysis and of construction, consists of parallel rectangular slats. For this configuration the results of Meyer et al. [1978] are relevant. They can be stated in terms of the Nusselt number for the convective heat transfer:
where RaL is the Rayleigh number corresponding to the spacing L between absorber and cover. The constants Cj and C2 are shown in Fig. 9.8.1 as a function of tilt angle /3 of the collector and aspect ratio A. The aspect ratio is the ratio of the distance between slats and the slat width L. (The slats are aligned along the horizontal direction.) Convection suppression is complete
Figure 9.8.1 Coefficients C, and C2 for use in Eq. (9.8.1) (From Meyer et al. [1978], as adapted by Duffie and Beckman [1980]; reprinted by permission of John Wiley & Sons).
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Active Solar Collectors and Their Applications
if Nu = 1. The variation of the coefficient C, in Fig. 9.8.1 gives a good indication of what can happen if slats are added to a flat plate collector. An aspect ratio A = oo corresponds to a collector without slats. As more slats are added, the aspect ratio decreases. The coefficient Ch and with it the Nusselt number, stays more or less constant until about A « 5. Between A « 5 and A ~ 0.5 the convection is actually enhanced relative to a collector without slats. Only for A < 0.5 is the convection reduced by the addition of slats. Hence one must be careful in the design of honeycombs if they are to be effective. Transpired collectors work on the principle of drawing the heat transfer fluid through small openings in the absorber. The cold fluid enters through the front of the absorber (i.e., the side facing the solar radiation). As the fluid absorbs heat, it moves into the absorber, thus keeping the front of the absorber cool. The absorber can be made of porous material, wire mesh, slitand-expanded metal foil, etc. Transpired absorbers seem to be used only for air collectors (in liquid collectors the relationship between conductivity and specific heat would imply too high a thermal mass to be practical). Compared to ordinary air collectors, transpired collectors are most effective in applications where the inlet temperature is low and the outlet temperature is high. This is likely to be the case in some industrial process heat applications. Fairly high pumping power requirements have been reported for some transpired collectors; see for instance Rhee and Edwards [1981]. On the other hand, in industrial air heating applications without storage (See Section 12.3) the flow rate can be so low that this point need not cause a problem. 9.9 HEAT LOSS COEFFICIENTS OF TYPICAL COLLECTORS
In this section the formulas of this chapter are applied to some typical collector configurations. Sample calculations are presented, and results are shown as graphs or curve fits. Once the optical efficiency ij0 and the U value have been calculated, the instantaneous collector efficiency is readily found from
9.9.1
Evacuated tubes
The heat flow from absorber to cover is given by Eq. (9.3.1):
where A^ = IT dl is the absorber surface area. The effective emissivity €eff is given by Eq. (9.3.2) because glass is a specular reflector of infrared radiation. For covers with «cov around 0.9 and selective coatings with eabs around O.I, ecff is very close to eabs, as demonstrated by the examples in Table 9.3.1. For
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237
example, with ecov = 0.88 and eabs = 0.1 the exact answer for eeff is 0.0987, only 1.3% smaller than eabs. In fact, the emissivities are rarely known with sufficient accuracy to be able to distinguish such differences, and hence the approximation
will usually be acceptable. Note that Table 9.3.1 also includes entries for heat mirror coatings on the cover. The emissivity of selective absorber coatings increases with temperature. Over the range of interest a linear approximation for this temperature dependence is usually quite accurate
For example, the values <<1000C) = 0.15 and €(300°C) = 0.25 have been reported by Gee et al. [1980] for a black chrome coating used in parabolic troughs. For evacuated collectors with flat absorber plate inside a glass tube (configuration of Figs. 1.3.4a-1.3.4c) the effective emissivity is still given by Eq. (9.3.2) provided tcov is replaced by the effective cavity emittance of the halfcylindrical cavity formed by the portion of the tube that faces the absorber plate. From the graphs for cavity emittances in Sparrow and Cess [1978] one can estimate that the cavity emissivity of the cover is approximately 0.95. The numerical effect on €efr of this change is negligible, as shown by Table 9.3.1. Not at all negligible, however, is the radiative loss from the back of the absorber plate. The back surface will usually be bare metal, for example, copper, (unless a CPC is used to illuminate both faces of the absorber plate). Bare copper has an emissivity in the range 0.02-0.05, depending on surface roughness. The surface area of the back is larger than the front if the fluid conduit is on the back. The cover temperature is determined by the requirement that the heat flow from absorber to cover equal the flow from cover to ambient. Assuming the simple combined radiative and convective heat transfer coefficient of Eq. (9.2.4) for the latter, one obtains the equation
If rabs and ramb are specified, solving for rcov is tedious, and one will resort to iterations. Tmv will be much "closer to T"amb than to Tabs, because of the high resistance between absorber and cover. For the purpose of calculating the U value and efficiency curve of the collector, one can avoid iterations if one starts by specifying the heat loss rather than the absorber temperature. Then the flow from cover to ambient yields rcov, and with that the flow from absorber to cover yields rabs. Repeating this for several heat loss values, say, 50 W/m2, 100 W/m2, and 150 W/m2,
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Active Solar Collectors and Their Applications
one can map out the entire collector efficiency curve without iterations. This is illustrated in Example 9.9.1. (If the heat balance includes separate terms for convection and for radiation, iterations will be needed; however, in a collector with several nodes [e.g., double-glazed collectors], the number of iterations is much smaller if one starts the calculation by fixing (2loss instead of rabs.) EXAMPLE 9.9.1
An evacuated tubular collector of the Owens-Illinois or General Electric type, Figs. 1.3.4d-1.3.4f, has absorber tube diameter d^ = 4.5 cm, outer cover diameter dmv - 5 cm, and emissivity ecff = 0.06. Ambient temperature is 10°C and h3mb = 25 W/m 2 K. What is the heat loss coefficient C7abs based on absorber surface area? SOLUTION
Start with Then Eq. (9.9.5) yields the cover temperature (note
Having found 7rov one can solve the other part of Eq. (9.9.5) for 7!,bs:
The U value t/abs relative to absorber surface area is
Additional numbers are given below:
rcov["C]
rabs[°C]
f/abs [W/m2 K]
10
10.4
38.2
0.35
25
10.9
69.6
0.42
50
11.8
108.8
0.51
75
12.7
138.5
0.58
100
13.6
162.9
0.65
300
20.8
283.0
1.10
i/abs is seen to increase from 0.35 W/m 2 K at 7;hs = 40°C to 1.10 W/m 2 K at 7 ahs = 283°C (the latter corresponds approximately to stagnation under full sunshine without reflector enhancement).
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239
For consistency with the other formulas in this book we need the U value relative to aperture, not absorber area. This depends on the optical design of the collector. If only the front of the tubes were receiving solar radiation (i.e., if no reflectors are used) then the aperture area A would be the diameter times the length, while the heat loosing absorber surface Aabs is the entire circumference times the length. Then the energy output
can be characterized by the usual equation for the collector efficiency r? =
if one takes the U value relative to aperture area to be
The optical efficiency for this collector is given by Eq.(5.2.20)
In practice one uses tubular collectors of the Dewar type only in combination with reflector enhancement, as discussed in Section 6.2 (See Fig 6.2. 1). Reflectors are desirable because they reduce the heat loss per aperture area and because reflectors are cheaper than tubes. If the reflector has a geometric concentration ratio4
then the U value relative to aperture area is
and the efficiency equation takes the usual form
The optical efficiency now includes the reflectivity p of the reflector
4
The concentration ratio of the above-mentioned arrangement without any reflectors is \/v.
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Active Solar Collectors and Their Applications
where (n) is the average number of reflections (See Section 6.1). («} is in the range 0.7-1.3 for typical low concentration reflectors. So far we have neglected conductive losses for evacuated collectors. Conduction in the tubes (at support points and at the neck) is small, but losses through the insulation of the manifold can be on the order of 50% of the radiative losses in the tubes. The conductive U value must be added to the radiative U value when calculating the collector performance. 9.9.2
Tubular receivers for parabolic troughs
Currently most parabolic trough collectors use tubular receivers with air between absorber and glazing. In that case the heat balance of Eq. (9.9.5) must be amended by the addition of a convective term on the left-hand side:
The convection heat transfer is given in Eq. (9.4.3). The simultaneous presence of convection and radiation precludes a closed form solution and one will resort to iterations. Numerical results are presented in Figs. 9.9.1-9.9.3, as heat loss coefficient Uabs relative to absorber surface area. These results for the reference design curve A are based on the following assumptions: ambient Tamb = 10°C absorber tube diameter 2.54 cm inner diameter of glazing chosen to minimize heat loss according to Fig. 9.9.1 (this corresponds to gap of approximately 0.7 cm and inner glazing diameter of approximately 3.9 cm) and absorber coating whose emissivity increases linearly with temperature, being 0.15 at 100°C and 0.25 at 300°C. Several improved receivers are also shown and labeled as follows in Fig. 9.9.3: B. emissivity decreased to 0.05 at 100°C and 0.15 at 300°C, still linear with temperature
Figure 9.9.1 Heat loss coeffiecient Vs.gap size for tubular receiver (From Geeet al.{1980] coefficient vs. gap size for tubular receier (From Gee
et al. [1980]).
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241
Figure 9.9.2 Heat loss coefficient variation with absorber diameter for tubular receiver (From Gee et al. [1980]). C. heat mirror coating with infrared reflectivity 0.15 added on inside of glazing D. air in annulus replaced by xenon E. annulus evacuated Figure 9.9.2 shows the effect of overall size on the loss coefficient [7abs. There is a slight variation because convection changes with scale. Figure 9.9.1 shows how (7abs varies with gap size for a fixed absorber diameter dabB = 2.54 cm. Figure 9.9.3 shows the variation with temperature. 9.9.3 Flat plates Given the formulas of Sections 9.2 and 9.3, calculating the heat loss of a flat plate collector is straightforward although somewhat tedious. At each node one equates incoming and outgoing heat flows, and then one solves for the temperatures by iteration. The calculation can be simplified greatly if one makes an intelligent guess for the temperatures of the nodes and linearizes the heat transfer equations by assuming constant resistances between the nodes. We illustrate this procedure with an example.
Figure 9.9.3 Heat loss coefficient vs. temperature for tubular receiver (From Geeetal. [1980]).
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Active Solar Collectors and Their Applications
EXAMPLE 9.9.2
Calculate the front loss coefficient of a single glazed flat plate collector with the following parameters: absorber to cover spacing 2.5 cm absorber emissivity eabs = 0.95 ambient air and sky temperature 7"amb = 10°C wind heat transfer coefficient /zwind absorber temperature rabs = 50°C collector tilt /? = 45° cover emittance ecov = 0.88 The frontal resistance l/f/ front is the sum of the resistances from absorber to cover and from cover to ambient. Using the symbols /zrad and /zconv for the radiative and convective conductances (as indicated in Fig. 9.1.2), one thus obtains
Let us guess an initial cover temperature of 20°C. Then the radiative conductances become
with
and
For the convective heat transfer between absorber and cover one can use Eq. (9.3.4) or Figs. 9.3.2 and 9.3.3. Using the latter we first obtain the correction factor jp, = 0.675 from Fig. 9.3.3 at an air temperature of 35°C. The appropriate value for the abscissa on the upper axis of Fig. 9.3.2 is then
for AT = 30°C and L = 25 mm. The corresponding Nusselt number is Nu = 3.0, which together with a conductivity of 0.027 W/m K for air yields the convective coefficient
Heat Transfer in Solar Collectors
243
Since the convective coefficient from cover to ambient was given as
we can now obtain the U value from Eq. (9.9.10) as
As a check on our guess for 7"cov we calculate the cover temperature corresponding to these heat transfer coefficients:
and find Tcm = 24.9°C. Using this value as a new cover temperature and repeating the calculation we obtain a new U value:
and a new cover temperature Tcm = 24.8°C. Even with a sizeable error in the initial guess for rcov the resulting U value is very close to the exact result. After the second iteration Tmv changes so little that there is no need to iterate any further. Front loss coefficients for a wide variety of designs and operating conditions are presented in Fig. 9.9.4. They can be summarized by the curve fit developed by S. A. Klein as reported in Beckman and Duffle [1980]. This fit reproduces t/front within ±0.3 W/m2 K for absorber temperatures between ambient and 200°C:
where N = number of covers;
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245
Figure 9.9.4 Front loss coefficient f/fron, for flat plate collectors of different designs under various operating conditions (From Duffie and Beckman [1980]; reprinted by permission of John Wiley & Sons).
/3 = «abs = «cov = rabs = ramb = ^wind =
collector tilt [degrees]; emissivity of absorber; emissivity of cover (= 0.88); mean absorber temperature [K]; ambient temperature [K]; convection coefficient from cover to ambient air [W/m2 K].
The variation of Ufronl with collector tilt is displayed in Fig. 9.9.5.
Figure 9.9.5 Dependence of C/from of flat plate on collector tilt 0 (From Duffie and Beckman [1980]; reprinted by permission of John Wiley & Sons).
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Active Solar Collectors and Their Applications
9.10 WARMING OF GLAZING BY ABSORPTION OF SOLAR RADIATION If the optical efficiency ?/0 is calculated from the optical properties of the collector materials and the U value as the thermal conductance from absorber to ambient, then one has not accounted for thermal effects of the solar radiation absorbed in the cover glazing. It is, however, possible to accommodate this eifect by a simple modification of TJO, provided one makes the usual approximation that the heat transfer coefficients are independent of temperature (errors from nonconstant U values are negligible in this case since we are dealing only with small shifts in temperature). For simplicity, let us consider a collector with a single glazing. It is sufficient to examine the linearized thermal network in Fig. 9.10.1 for the heat flows between absorber, cover, and ambient. Without absorption the cover temperature is rcov as shown in Fig. 9.10. la. Suppose now that the cover absorbs a fraction «cov of the incident solar radiation /, raising the cover temperature to T'mv while the absorber temperature rabs is kept fixed. The heat loss through the front of the collector is t/abs,cov(^abS - Tcm) without, and U^^T^ - T'mv) with, absorption, where C/abscov is the heat transfer coefficient from absorber to cover. The difference between these two heat loss terms
is a heat loss reduction which can be treated as an additional energy input into the absorber. If T?O and U have been calculated without taking absorption in the cover into account, then the true efficiency of the collector is
To find (T'cm — rcov) we consider the energy balance of the cover. Without absorption the network in Fig. 9.10. la yields
Figure 9.10.1 Thermal
network for front losses for a single-cover collector (a)
without and (b) with absorption in the cover.
Heat Transfer in Solar Collectors
247
With absorption Fig. 9.10.1b is relevant; it yields
Subcontracting the last two equations from each other we find
Solving for T'cm — rcov and substituting it into Eq. (9.10.1) produces the heat loss difference in the form
This can be rewritten in terms of the front loss coefficient
as
It is convenient to introduce the term "effective optical efficiency" for the combination of TJO and A0/7
Hence, the effective optical efficiency for a single glazed collector is
For a flat plate collector that has several covers of different materials, Duffie and Beckman [1980] have shown that the effective optical efficiency is given by the equation
where «cov, = absorptance of zth cover, T, = transmittance of the cover systems above the (z + l)st cover, a, = coefficients in Table 9.10.1.
248
Active Solar Collectors and Their Applications TABLE 9.10.1
Constants for Use in Eq. (9.10.11 )a
Covers
a, eabs = 0.95
1 2
a, a, a2 a, a2
3
fl3
0.27 0.15 0.62 0.14 0.45 0.75
(abs = 0.10
0.21 0.12 0.53 0.08 0.40 0.67
0.13 0.09 0.40 0.06 0.31 0.53
"From Duffie and Beckman [1980]. Reprinted by permission of John Wiley & Sons, Inc.
The values of a, depend on temperature and emittance of the absorber as well as on ambient conditions. The values in Table 9.10.1 were calculated for absorber emittances of 0.95, 0.5, and 0.1, for Tabs = 100°C, Tamb = Tsky = 10°C and for a convective heat transfer coefficient from outer cover to ambient of 24 W/m2 °C. The angular dependence of TJO efr can be calculated by including the angular dependences of ??0, acov/, and r,. In practice, there is usually no need to calculate the correction terms in Eq. (9.10.11) with great precision since the correction is small and an error in estimating the correction has relatively little effect on the useful energy delivered by a collector. For example, a nonselective flat plate collector with one cover has an a, = 0.27. If the cover absorbs acov, = 0.04 of the incident radiation, then the correction term is a covl a, = 0.01, in other words, ?;oefr is on the order of 1% greater than TJO. A 10% error in estimating at will cause an error on the order of 0.1% in the delivered energy. In many focusing collectors (e.g., parabolic troughs), the glazing will be made of borosilicate glass in order to withstand the high temperature close to the absorber; this type of glass absorbs so little solar radiation that the difference between r;0 eff and 770 is negligible.
REFERENCES Abdel-Khalik, S. I. et al. 1978. "Natural Convection in Compound Parabolic Concentrators—A Finite-Element Solution." J. Heat Transfer 100:199. ASHRAE. 1977. Handbook of Fundamentals, Chapter 22, Table 1. Atlanta, GA: American Society of Heating, Refrigeration and Air Conditioning Engineers. Buchberg, H. and Edwards, D. K. 1976. "Design Considerations for Solar Collection with Cylindrical Glass Honeycombs," Solar Energy 18:193. Buchberg, H., Catton, I., and Edwards, D. K. 1976. "Natural Convection in Enclosed Spaces—A Review of Application to Solar Energy Collection." J. Heat Transfer 98:182. Cooper, P. I. 1981. "The Effect of Inclination on the Heat Loss From Flat Plate Collectors." Solar Energy 27:413. Duffie, J. A. and Beckman, W. A. 1980. Solar Engineering of Thermal Processes. New York: Wiley Interscience.
Heat Transfer in Solar Collectors
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Felland, J. R. and Edwards, D. K. 1978. "Solar and Infrared Radiation Properties of Parallel-Plate Honeycomb." J. Energy 2:309. Francia, G. 1961. "A New Collector of Solar Radiant Energy—Theory and Experimental Verification." UN Conference on New Sources of Energy, p. 554. Fujii, T. and Imura, H. 1972. Int. J. Heat Mass Transfer Vol. 15. Gee, R., Gaul, H. W., Kearney, D., and Rabl, A. 1980. "Long Term Average Performance Benefits of Parabolic Trough Improvements." Report SERI/TR-632-439. Golden, CO: Solar Energy Research Institute. Hollands, K. G. T. 1965. "Honeycomb Devices in Hat Plate Solar Collectors." Solar Energy 9:159. Hollands, K. G. T. et al. 1976. "Studies on Methods of Reducing Heat Losses from Flat Plate Collectors." Report for ERDA Contract EY-16-G-02-2597. Ontario, Canada: Dept. of Mechanical Engineering, University of Waterloo. Hollands, K. G. T., Unny, T. E., Raithby, G. D., and Konicek, L. 1976. "Free Convection Heat Transfer Across Inclined Air Layers." Trans. Amer. Soc. Mech. Eng., J.Heat Transfer 98:189. Holman, J. 1976. Heat Transfer. New York: McGraw-Hill. Kreith, F. 1973. Principles of Heat Transfer, 3rd ed. New York: Intext Educational Publishers. Kreith, F. and Kreider, J. F. 1980. "Principles of Thermodynamics and Heat Transfer Applied to Solar Energy." In Solar Energy Handbook, Kreider, J. F. and Kreith, F., editors. New York: McGraw-Hill. Kuehn, T. H. and Goldstein, R. G. 1978. "An Experimental Study of Natural Convection Heat Transfer in Concentric and Excentric Horizontal Cylindrical Annuli." J. Heat Transfer 100:635. Malhotra, A., Garg, H. P., and Rani, U. 1980. "Minimizing Convective Heat Losses in Flat Plate Solar Collectors." Solar Energy 25:521. McAdams, W. H. 1954. Heat Transmission, 3rd ed. New York: McGraw-Hill. Meyer, B. A. et al. 1978. "Natural Convection Heat Transfer in Small and Moderate Aspect Ratio Enclosures—An Application to Flat-Plate Collectors." In Thermal Storage and Heat Transfer in Solar Energy Systems, Kreith, F., Boehm, R., Mitchell, J., and Bannerol, R., editors. New York: ASME. Rabl, A. 1976. "Optical and Thermal Properties of Compound Parabolic Concentrators." Solar Energy 18:497. Rabl, A., O'Gallagher, J., and Winston, R. 1980. "Design and Test of Non-Evacuated Solar Collectors With Compound Parabolic Concentrators." Solar Energy 25:335. Raithby, G. D., Hollands, G. T., and Unny, T. R. 1977. "Analysis of Heat Transfer by Natural Convection Across Vertical Fluid Layers." Trans. Amer. Soc. Mech. Eng., J. Heat Transfer 99:287. Rhee, S. J. and Edwards, D. K. 1981. "Comparison of Test Results for Flat Plate, Transpired Flat Plate, Corrugated, and Transpired Corrugated Solar Air Heaters." Paper No. 81-WA/SOL-18. ASME Winter Annual Meeting, Washington, D.C. Sparrow, E. M. and Tien, K. K. 1977. "Forced Convection Heat Transfer at an Inclined and Yawed Square Plate-Application to Solar Collectors." ASME J. Heat Transfer 99:507. Sparrow, E. M. and Cess, R. D. 1978. Radiation Heat Transfer. New York: Hemisphere Publishing Corp., McGraw-Hill Book Co. Sparrow, E. M., Ramsey, J. W., and Mass, A. M. 1979. "Effect of Finite Width on Heat Transfer and Fluid Flow about an Inclined Rectangular Plate." /. Heat Transfer 101:199.
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Active Solar Collectors and Their Applications
Sparrow, E. M., Nelson, J. S., and Tao, W. Q. 1982. "Effect of Leeward Orientation, Adiabatic Framing Surfaces, and Eaves on Solar Collector-Related Heat Transfer Coefficients." Solar Energy 29:33. Tabor, H. 1958. "Radiation, Convection and Conduction Coefficients in Solar Collectors." Bull. Res. Council Israel 6C:155. U.S. Home Finance Agency. 1954. "The Thermal Insulating Value of Airspaces." Report #32. Washington, D.C.: US Government Printing Office.
10. HEAT TRANSFER FACTORS
10.1
TEMPERATURE DISTRIBUTION IN COLLECTOR
In practical applications one needs to know how the efficiency of a collector depends on the temperature of the heat transfer fluid flowing through it, whereas the temperature of the absorber surface is of little interest. Hence, it is desirable to replace Eq. (9.9.1) by one that refers directly to the fluid. There are several possibilities for specifying the temperature of the fluid. Of particular interest are the temperature T-m of the fluid at the collector inlet, the fluid temperature rou, at the collector outlet, and the mean fluid temperature
One can also base the collector efficiency on the temperature at which heat can be delivered across a heat exchanger. As shown in the following paragraphs, the basic linear form of Eq. (9.9.1) remains the same; only a straightforward modification of the collector parameters is needed to account for a different temperature base, namely, multiplication by the appropriate heat transfer factor. In the previous section we talked about the absorber temperature as if it were constant over the entire absorber surface. In order to derive the heat transfer factors, one needs to take a closer look at the actual temperature distribution in the collector. Figure 10.1.1 illustrates a typical pattern in a flat plate collector of the plate and tube type. Solar radiation is absorbed over the entire absorber surface and then conducted through the absorber plate to the heat transfer fluid. Therefore, the absorber must be hotter at points that are further from the tubes. As the fluid advances through the collector, it warms up; hence, the temperature of the absorber plate must also increase in the flow direction. The resulting temperature distribution is sketched schematically in Fig. 10. Lib as a function of the distance y along the flow direction, in Fig. lO.l.lc as a function of the distance x perpendicular to the flow direction, and in Fig. 10.1. Id as a function of both x and y. An exact analysis would require two-dimensional (and, in the case of thick absorbers, three-dimensional) heat transfer calculations—a tremendous complication without noticeable benefit. In practice one makes the approximation that heat conduction in the x direction, i.e., parallel to the fluid 251
252
Active Solar Collectors and Their Applications
Figure 10.1.1 Temperature distribution (qualitative) in the absorber of a flat plate collector, (a) Schematic diagram of absorber, (b) Temperature distribution along flow direction, (c) Temperature distribution perpendicular to flow direction, (d) Temperature distribution in the absorber plate. (From Kreith and Kreider [1978]).
flow, has no effect on the collector performance. With this approximation the analysis separates into two manageable pieces: (i) the heat transfer in the direction perpendicular to the flow, discussed in Section 10.2, and (ii) the temperature increase along the flow direction, discussed in Section 10.3. The correction factor for heat exchangers is derived in Section 10.4. Section 10.5 shows how the efficiency equation of a single collector module can be modified to include the heat losses from piping between collectors and point of use in a collector array. Further heat transfer factors could be defined. For example, heat transfer factors for steam producing solar energy systems have been calculated by Gee [1984].
10.2 MEAN FLUID TEMPERATURE In most collectors one can analyze the heat transfer from absorber surface to fluid by the following argument. Consider a cross section of the collector perpendicular to the flow direction of the fluid. The mean absorber temperature in this section is Tabs while the mean fluid temperature is Tm. Accord-
Heat Transfer Factors
253
ing to Eq. (9.9.1) a collector delivers heat in the amount (per aperture area)
when the absorber temperature is Tabs. This heat must be conducted from the receiver surface into the heat transfer fluid which is at a fluid temperature Tm. Hence q must equal
where f/abs, /-is the thermal conductance from absorber surface to fluid. Since we want to eliminate 7"abs in favor of Tm we solve Eq. (10.2.1) for T^
and insert this expression for Tabs into Eq. (9.9.1):
Collecting q on the left side, one obtains
If one defines a heat transfer factor Fm as
then the collector efficiency equation takes the simple form
By contrast to some of the efficiency equations to be derived in the following sections, the equation based on Tm is exact even if [/is not constant. Rewriting Fm in the form
254
Active Solar Collectors and Their Applications
we recognize Fm as the ratio of the resistance from absorber to ambient and the resistance from fluid to ambient. In the older literature it has frequently been designated by the symbol F and called the "collector efficiency factor." In view of the large number of similar factors that can be derived for different temperature bases it is more logical to use the generic name "heat transfer factor" and the symbol F with the appropriate mnemonic subscript, as recommended by Lunde [1980]. Fm is a parameter that depends on the design of the collector and, to a much smaller degree, on the flow rate. In practice, one can usually approximate it by a constant value even when the flow rate varies. For most collectors Fm lies in the range of 0.8-1.0, being smaller for air collectors and larger for liquid collectors. The following examples will illustrate how Fm is calculated and what values it takes for typical collectors. EXAMPLE 10.2.1 A flat plate collector has an absorber plate which consists of two parallel plates of steel which are sealed together at the edges so that water can flow between the plates. (To prevent bulging under water pressure, the plates are joined not only at the edges but also at several intermediate points; e.g., by adding dimples to the plates and welding them together where the dimples touch.) The collected heat must be conducted from the absorber surface through the upper steel plate into the water flowing between the plates. The resistance l/f/ abs- /from the absorber surface to the water is the sum of the resistance across the upper plate and across the fluid film:
where t = thickness of steel plate (equals 1 mm) k - conductivity of steel (50 W/m) h = convective film coefficient for water (equals 500 W/m2 °C) (For typical values of h see Table 9.1.1.) The numbers in parentheses are typical values and indicate that the resistance across the fluid film is about two orders of magnitude larger than the resistance across the steel plate; hence we can approximate
If the collector has a U value of 5 W/m 2 °C the heat transfer factor is
very close to unity.
Heat Transfer Factors
255
EXAMPLE 10.2.2
Suppose the collector of Example 10.2.1 were used with air instead of water. (In practice one would not use the same collector with water and with air, because of problems with pumping power, heat transfer, or thermal inertia. In the present case one would space the two steel plates further apart if used with air, and no interior joints would be needed to keep the two plates together. Such design changes do not affect the calculation of Fm.) Typical convective heat transfer coefficients for forced convection of air in a solar collector are on the order of h = 20 W/m2 °C. Now the resistance across the steel plate is even more negligible, and with £/abs, f = /z the heat transfer factor turns out to be
for the same U value. In air collectors the resistance between absorber and fluid can cause a severe penalty which should be minimized by careful design. EXAMPLE 10.2.3
A parabolic trough collector has a tubular receiver, and oil is used as heat transfer fluid with a convective heat transfer coefficient of h = 100 W/m2 °C. The geometric concentration ratio is C = 25 and the U value U = 0.5 W/m2 K, typical of such collectors if the absorber is selective and enclosed in a nonevacuated glass tube operating around 200-300°C. Of course, the U value is based on the aperture area A of the collector, and for consistency with Eq. (10.2.6) [7abs, f must also be based on A. The coefficient h, on the other hand, refers to the surface area of the inside of the receiver tube. For simplicity, let us assume that the tube wall is thin compared to its diameter, and let us also neglect any radial or circumferential temperature gradients in the tube wall. Then we can write Eq. (10.2.1) in terms of h as
where ^4abs = A/C is the absorber tube surface area and C the concentration ratio. Hence, t/abs,, is
and the heat transfer factor turns out to be
It is approximately independent of concentration ratio, because the U value is also inversely proportional to C.
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Active Solar Collectors and Their Applications
With air instead of oil, the heat transfer coefficient h would be on the order of 20 W/m 2 K and C/abs, f = 0.8 W/m2 K, which results in a rather low value Fm = 0.62 for the heat transfer factor. However, if the receiver were evacuated with a selective coating of low emittance, Fm could be quite close to unity, as shown by the next example. EXAMPLE 10.2.4
An evacuated tubular collector of the Owens-Illinois type, see Fig. 1.3.4e, uses air as heat transfer fluid with a film coefficient h = 20 W/m 2 K. The effective emittance of the selective absorber is e = 0.07 and the corresponding U value relative to absorber surface area is approximately C/lubc = 0.6 W/m 2 K. As long as both h and C/tube refer to the same surface area (and again we neglect the difference between inside and outside surface areas of the tube), one can insert them directly into Eq. (10.2.5), with the result
A comment should be added about the [/value to be used in Eq. (10.2.5). Since Fm accounts for the temperature difference between absorber surface and fluid, the U value in Eq. (10.2.5) is the heat loss coefficient from the absorber surface to ambient. In most collectors, with the exception of evacuated tubular collectors, this heat loss coefficient is close to the overall U value of the collector because losses from the collector manifold are relatively small. In evacuated tubular collectors, on the other hand, losses from the manifold can be almost as large as losses from the absorber surface, and one should not use the overall collector loss coefficient in Eq. (10.2.5). EXAMPLE 10.2.5
As a final example, consider the important case of a flat plate collector where the water flows in tubes that are bonded to an absorber plate, as shown in Fig. 10.2. la. From the numbers in Example 10.2.1, we have learned that the resistance across the tube wall is negligible compared to the film coefficient of the tube. Furthermore, the tube wall can be considered isothermal. Let us assume that the tube is welded to the plate [if it is not, one needs to add the resistance across the bond to the right side of Eq. (10.2.16)]. Since the temperature varies along the x direction of the plate, it is convenient to start with Eq. (10.2.7) as applied to a strip of the collector of width 2w + b and length L that contains one tube at its center. Then the numerator of Eq. (10.2.7) is the resistance
from absorber plate to ambient while the denominator is the resistance Rf#mb from fluid to ambient. The latter is the sum
Figure 10.2.1 Flat plate collector with sheet and tube design, (a) Geometry (b) Temperature distribution if plate heated by sun. (c) Temperature distribution if tubes are losing heat to ambient.
257
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Active Solar Collectors and Their Applications
where
where D is the interior tube diameter, and -ftu.be, amb is the resistance from tube wall to ambient. The latter can be calculated by the following method. Consider a situation where the collector is hot, but no sun is shining. If the temperature of the tube wall and hence at the base of the tube is Tb, then the heat flow in the direction from tube wall to ambient is
This heat flow can be written as a sum of two terms
where k = conductivity of the plate and t = thickness of the plate. The first term is the heat flow from the base directly to ambient while the second represents the flow from the tube into the plate between tubes. It can flow to the right and to the left portion of the plate and, hence, the factor of 2. Calculating the temperature gradient for the second term is a common cooling fin problem [Kreith, 1973]. Typical temperature distributions along the x direction of the absorber plate are shown in Fig. 10.2.1b for the case when absorbed solar heat is conducted to the tubes, and in Fig. 10.2.1c for the case when the absorber is hot but no sunlight is incident. The resistance -fttube, amb is, of course, independent of solar radiation and can be calculated for the simpler case of Fig. 10.2.1c, as assumed already in Eqs. (10.2.18) and (10.2.19). To find
consider the heat balance of a narrow strip of the fin having width Ax and length L. By energy conservation the heat loss from this strip, L&xU [T(x) — Tamb] must equal the difference between the heat flow
Heat Transfer Factors
259
into the strip at x + Ax and the heat outflow at x
This yields the equation
Taking the limit Ax -» 0 one obtains the differential equation
with
Its most general solution is
The integration constants c, and c2 are determined by the boundary conditions
and
(The latter follows from symmetry about x = 0.) This fixes the solution as
and yields the temperature gradient at the base as
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Active Solar Collectors and Their Applications
A useful concept for the analysis of cooling fins is the fin efficiency, defined as ratio of the heat which is transferred through a real fin over the heat which would be transferred through a fin of infinite conductivity. For the present case of a fin of uniform thickness, the fin efficiency is given by
This quantity is plotted in Fig. 10.2.2 as a function of mw. In terras of F, the temperature gradient at the base can be written as
Returning to Eq. (10.2.19) we now obtain the following expression for the heat flow from tube to ambient:
Combining Eq. (10.2.30) with Eqs. (10.2.16) through (10.2.18) yields the resistance from fluid to ambient:
Figure 10.2.2 Fin efficiency for tube and sheet solar collectors (Adapted from Duffie and Beckman [1980]; reprinted by permission of John Wiley & Sons).
Heat Transfer Factors
261
Finally, we recall the definition of Fm as the ratio of the resistance from absorber surface to ambient, Eq. (10.2.15), and the resistance from fluid to ambient, Eq. (10.2.31), to obtain the heat transfer factor for this collector:
Numerical values corresponding to this equation can be read from Fig. 10.2.3 for various values of the design parameters. Even though Eq. (10.2.32) was derived for plates of uniform thickness, it can be directly applied to tapered plates, provided one uses for the fin efficiency the equation corresponding to tapered plates [Kreith, 1973]. Such plates (i.e., plates which are thicker near and thinner far from the tubes) have been suggested as a means of reducing the mass and thereby the cost of absorber plates without sacrificing performance. In some flat plate absorbers there is a significant resistance across the bond between tube and plate. In that case the equation for Fm is augmented
(a)
Figure 10.2.3 Heat transfer factor Fm versus tube spacing 2w for b = 10mm diameter tubes for the configuration of Fig. 10.2.1. Curves are labeled according to kt where k = conductivity and t = thickness of plate. Different graphs correspond to different collector U values and heat transfer coefficients h between fluid and tube, (a) h = 100 W/m2 °C; (b) h = 300 W/m2 °C; (c) h = 1000 W/m2 °C. (Adapted from Duffie and Beckman [1980]; reprinted with permission of John Wiley & Sons).
Figure 10.2.3 262
(continued)
Heat Transfer Factors
263
by a term for this bond resistance:
where kb = thermal conductivity 1 tb = thickness I of bond material. b = width J
10.3
FLUID INLET TEMPERATURE
In order to base the efficiency equation on the fluid inlet temperature T-m, one needs to calculate the temperature rise along the flow direction. Consider a short section of a collector, having length Ay (along flow direction) and width w (perpendicular to flow direction), and let m be the flow rate and c the specific heat of the fluid. The length Ay is short enough that the fluid temperature in the efficiency equation can be treated as constant. Then the net input of solar energy into the fluid between y and y + Ay is, by Eq. (10.2.6),
By energy conservation the fluid temperature must increase from T(y) to T(y + Ay) = T(y) + AT according to
Combining these equations and taking the limit Ay -» 0 one obtains the differential equation
If U is constant, then the solution is
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Active Solar Collectors and Their Applications
and the constant is fixed by the boundary condition T(y = 0) = 7"in. Hence the temperature at point y is given by
In particular, for y = L = length of collector, we have T(y = L) = Toui and wL = A = aperture area of collector, and thus the outlet temperature is given by
If the entire absorber surface were at the fluid inlet temperature, then the energy gain would be AfaJ - U(T-m - Tamb)]. The real energy gain is mc(TOM — Tin). It is convenient to define a heat transfer factor Fin as the ratio of these two quantities:
With this definition the efficiency equation ?? = mc(ToM — Tia)/AI can be written as
This has the same form as the efficiency equations based on absorber temperature, Eq. (9.9.1), or mean fluid temperature, Eq. (10.2.6). An explicit expression for Fin is readily found after a few straightforward modifications of Eq. (10.3.7):
Heat Transfer Factors
265
Inserting Ea. (10.3.6) we find
Numerical values for F,n can be presented in a single graph ifF,n/Fm is plotted versus
as in Fig. 10.3.1. The largest value of Fm is Fm, corresponding to such high flow rates that the entire collector is essentially at the inlet temperature. As the flow rate decreases, the average collector temperature increases; Fin accounts for the resulting drop in efficiency. For very low flow rates, Fin approaches zero because the collector reaches stagnation conditions and does not deliver any useful energy. Equation (10.3.8) (sometimes referred to as Hottel-Whillier-Bliss equation) is an extremely useful tool for the analysis of systems with active solar collectors. The fluid inlet temperature is usually known directly in terms of the instantaneous values of the system parameters, whereas the mean fluid temperature is not known directly because it also depends on the instantaneous insolation. Equation (10.3.8) bypasses the need to calculate the mean fluid temperature explicitly. Equation (10.2.6) for the efficiency in terms of the mean fluid temperature rm was derived on the assumption that the collector was isothermal along the flow direction; in other words, that the temperature rise from inlet to outlet was negligible. Having taken the temperature rise into account by
Figure 10.3.1 Heat transfer factor Fin as ratio Fin as ratio FJF,n versus mc/(A UFm) (From Duffie and Beckman [1980]. Reprinted with permission of John Wiley & Sons).
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Active Solar Collectors and Their Applications
means of the heat transfer factor Fin we can now return to Eq. (10.2.6) and examine its validity for realistic temperature differences. The collector output Q is related to temperature rise and flow rate by
Hence the arithmetic mean fluid temperature Tm = (7"in + rout)/2 can be written as
When this is inserted into Eq. (10.2.6) we obtain a result for the efficiency
which is approximate for nonzero temperature rise. (Or to put it another way, the true average temperature is different from Tm.) Let us rearrange this eauation in the form
because the term in brackets is the exact efficiency rj according to Eq. (10.3.8). The ratio of the approximate and the exact efficiency is
It depends only on the single variable FmUA/(mc), as can be seen by inserting Eq. (10.3.9) for Fm:
It is unity at FmUA/(mc) = 0 as it should be, and it increases monotonically with FmUA/(mc). The difference between 77approx and rj is most pronounced at low flow rates and high U values. In practice the most extreme case is probably one with Fm U = 8 W/m2 K and water at m/A = 2 gm/m2 sec. In this 0.5 the difference between r;approx and i\ is only 2%, and for FmUA/(mc) = 0.25 it becomes entirely negligible, as shown explicitly in the following example.
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267
EXAMPLE 10.3.1
A flat plate collector with Fmr?0 = 0.80 and FmU = 5.0 W/m2 °C has aperture area A - 2.0 m2 and is used under the following conditions:
Calculate the efficiency exactly (i.e., using Eq. (10.3.9) for Fin, which accounts for the nonlinear temperature rise) and compare the result with the efficiency as obtained from Eq. (10.2.6) with the simple arithmetic mean temperature. SOLUTION
Evaluating Eq. (10.3.9) we find
Fmf)0 can now be obtained from
Since Tw = Tamb, Eq. (10.3.8) reduces to
The output of the collector
is related to the outlet temperature via
Solving for Toul we find
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Active Solar Collectors and Their Applications
The simple arithmetic mean temperature is
If we use this value of Tm in Eq. (10.2.6) we obtain a slightly different result for the efficiency
The difference between this number and the exact result, 0.712, is indeed negligible. We conclude this section with some comments about the validity of the Hottel-Whillier-Bliss (HWB) equation. First of all, the equation applies to concentrating collectors as well as to flat plates. Second, the derivation of Eq. (10.3.8) required the assumption that U is constant; hence, the HWB equation is less exact than Eq. (10.2.6), which is based on the mean fluid temperature. Third, Eq. (10.3.8) does not apply to collectors with a thermal short circuit between inlet and outlet. Some evacuated tubular collectors, for example, the one in Fig. 1.3.4e, fall into that category. Here the fluid enters through the center tube and returns through the annulus between center tube and absorber tube. Some of the heat absorbed by the fluid in the annulus is conducted to the inner tube, thus raising the temperature of the fluid before it even comes in contact with the absorber tube. This temperature increase reduces the efficiency below the value predicted by naive application of Eqs. (10.3.8) and (10.3.9). This penalty is acceptable in evacuated collectors because of their low heat loss coefficient. For a detailed analysis of this effect the reader is referred to Thodos [1976]. 10.4 HEAT EXCHANGER FACTOR The combination of a solar collector with a heat exchanger shown in Fig. 10.4.1 can be described by a linear efficiency equation with the standard parameters ?;„ and U provided one multiplies the equation by the appropriate correction factor [de Winter, 1975]. Thus, it becomes possible to describe the combination of collector plus heat exchanger as if it were just a simple collector, without worrying about the temperatures Tin and rout in the collector-heat exchanger loop. To begin this analysis let us consider the heat flow across a heat exchanger. This is shown schematically in Fig. 10.4.2. The two fluid streams passing through the heat exchanger have either equal flow capacitance rates or one of the two rates is larger than the other. Hence we have labeled the properties of the fluid streams by subscripts s and / according to which is smaller and which is larger. Hence in the 5 stream the flow capacitance rate is (me), and inlet and outlet temperatures are Tsl and Tso. The inlet and out-
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Heat Transfer Factors
Figure 10.4.1 Schematic of liquid system using a collector heat exchanger between collector and load or tank.
let temperatures in the / stream are Tti and T,0 and the rate is (me}, with
Now, what is the largest amount of heat <2max that could possibly be transferred if the inlet temperatures and the flow capacitance rates are specified? The outlet temperatures must be bounded by the inlet temperatures (otherwise heat would flow spontaneously from cold to hot in clear violation of the second law of thermodynamics—since we are excluding friction or other mechanical effects). Hence Qmn corresponds to bringing the smaller stream to the temperature of the incoming larger stream:
This would require an infinite heat exchanger area. The heat flow Q actually achieved can be expressed as a fraction e of Qmm'.
(. is called heat exchanger effectiveness. It is a useful concept because it is independent of temperature level and depends only on the heat exchanger properties and the flows. For the most important heat exchanger types the equation for e is listed in Table 10.4.1. (For derivations the reader is referred to Eckart and Drake [1959] or Kays and London [1964].) For the counter-
Figure 10.4.2 Heat transfer in a heat exchanger.
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Active Solar Collectors and Their Applications
TABLE 10.4.1 Heat Exchanger Effectiveness for Several Heat Exchanger Types" Relation Flow geometry Double pipe Parallel flow Counterflow Cross flow Both fluids unmixed Both fluids mixed CmM mixed, Cmin unmixed Cmax unmixed, Cmin mixed Shell and tube One shell pass, 2, 4, 6 tube passes a
From Kreith and Kreider [1980].
current heat exchanger the eifectiveness is plotted in Fig. 10.4.3. For liquidto-liquid heat exchangers in solar applications Kreider and Kreith [1982] recommend values o f « i n the range of 0.7-0.8 as economical optimum. To derive the heat exchanger factor, we first write the power output Q of the collector as
Figure 10.4.3 Heat exchanger effectiveness ( versus number of transfer units A' = (UA)HX/(mc)m,n and flow rate ratio C = (wf) min /(mc) max for counterflow heat exchanger (Adapted from Krcith and Kreider[1980]).
Heat Transfer Factors
211
Of course, Q equals also
where (mc)con is the flow rate-heat capacitance in the collector loop. Solving the second equation for Tin and inserting this expression for T-m into the first equation, one can rewrite Q in the form
with the heat transfer factor
Assuming zero heat loss in the piping between collector and heat exchanger, the collector outlet temperature is one of the inlet temperatures of the heat exchanger; the other is THX in. The heat flow Q from the collector equals the heat flow across the heat exchanger; the latter can be expressed in terms of the effectiveness e as
where
Now we can solve Eq. (10.4.8) for TOM and insert this value for rou, into Eq. (10.4.6). Rearranging terms and isolating Q on the left-hand side one finds
with a new heat transfer factor, the heat exchanger factor Fx:
This equation is analogous to the other equations for collector efficiency as collector output. It is very useful because with its help one can treat the collector-heat exchanger combination as if it were a simple collector with efficiency equation
Numerical values for Fx as a function of effectiveness e, flow rates, and (AFmU) are shown in Fig. 10.4.4.
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Active Solar Collectors and Their Applications
Figure 10.4.4 Collector heat exchanger correction factor as a function of t (mc)mm/ (mc)coii and (mc)con/(F,nUA) (Adapted from Duffie and Beckman [1980]. Reprinted with permission of John Wiley & Sons). EXAMPLE 10.4.1 A collector with A = 2 m2 and Fin??0 = 0.75 and Fin U = 4.0 W/m2 °C is used with a heat exchanger. In the collector loop a glycol solution with c = 3.3 kJ/kg °C is circulated at a flow rate of m = 0.03 kg/sec, and the value of-Fin corresponds to this flow rate-heat capacity product. (Note: if test results for Fin?;0 and Fin U were obtained with a different me, then Fia must be recalculated first, using the formulas of Section 4.2.) On the load side of the heat exchanger, water (c = 4.186 kJ/kg °C) flows at a rate m = 0.03 kg/sec. The heat exchanger is of the counterflow type with (UA)HX = 0.2 kW/°C. What is the heat delivered across the heat exchanger if rHX in = ramb = 10°C and 7 = 1000 W/m2? SOLUTION In the collector loop (mc)cotl = 99.0 W/°C, in the load loop (mc)M = 125.58 W/°C; hence (mc)min = 99.0 W/°C and (mc)max = 125.58 W/°C. First find the heat exchanger effectiveness from Table 10.4.1 for counterflow:
with
and
This yields a heat exchanger effectiveness
Heat Transfer Factors
273
From Eq. (10.4.11) we find
At r HX ,in = Tamb the heat loss term in Eq. (10.4.12) drops out and the heat delivered to the load is
10.5
COLLECTOR ARRAYS AND PIPE LOSS FACTORS
In most applications several collector modules will be interconnected to form a collector field or array. If heat losses from pipes between modules are negligible, then the efficiency curve of a collector array is identical to that for a single module. In particular, the heat transfer factor F-m of the array
depends only on the ratio of array flow rate marray and array area Atmy, not on array layout (i.e., series, parallel, or some combination of series and parallel).1 This follows because the derivation of the heat transfer factor Fin depends only on the ratio of area and flow rate. The energy lost from pipes or ducts between collector and point of use can be significant, typical values being in the range of 1% to 10%. Beckman [1978] has shown how to treat the combination of collector and pipes as if it were a simple collector. To derive the necessary correction factors for the collector parameters, it is helpful to consider the temperature distribution along pipes and collector as sketched in Fig. 10.5.1. Fluid enters the pipe at Tin and due to heat losses its temperature is reduced by an amount ATj n by the time it enters the collector. In the collector, the fluid is heated to the outlet temperature of the collector. By the time it reaches the end of the outlet pipe, the fluid has cooled off to Toul. The power delivered by the collector-pipe combination is
'Some rather complicated equations have been published in the literature to calculate the efficiency of several collectors in series. Such equations are not necessary if one uses Eq. (10.5.1).
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Active Solar Collectors and Their Applications
Figure 10.5.1 Temperature distribution through a ductcollector system (Adapted from Beckman[ 1978]).
It is related to the collector parameters by the equation
If the surface areas of the inlet and otlet pipes are A and A and if theri heat loss coefficient is U [ in W/m C], then the pipe loss can be approximated by
(assuming that the temperature drops along the pipes are relatively small). Using Eq. (10.5.2) to eliminate Tout, we can rewrite the last equation as
The decrease A7"in on the inlet side can be approximated by
Substituting the last two equations into Eq. (10.5.3) and rearranging terms one can express the delivered power Q as
Heat Transfer Factors
275
This looks just like the usual collector equation
if one defines the parameters ri'0 and U for the collector plus pipe combination as
and
If a system contains both pipes and heat exchanger, one first modifies the collector parameters according to Eqs. (10.5.9) and (10.5.10) and then uses the modified U value U in Eq. (10.4.11) for the calculation of the heat exchanger factor Fx.
REFERENCES Beckman, W. A. 1978. "Duct and Pipe Losses in Solar Energy Systems." Solar Energy 21:531. Bliss, R. W. 1959. "The Derivations of Several 'Plate Efficiency Factors' Useful in the Design of Flat-Plate Solar-Heat Collectors." Solar Energy 3:55. de Winter, F. 1975. "Heat Exchanger Penalties in Double-Loop Solar Water Heating Systems." Solar Energy 17:335. Duffie, J. A. and Beckman, W. A. 1980. Solar Engineering of Thermal Processes. New York: John Wiley & Sons. Eckart, E. R. G. and Drake, R. M. 1959. Heat and Mass Transfer, 2nd ed., New York: McGraw-Hill. Gee, R. 1984. "Simple Heat Exchange Factors for Steam Producing Solar Systems." ASME J. Solar Energy Eng. 106:364. Hottel, H. C. and Whillier, A. 1958. "Evaluation of Flat Plate Collector Performance." Transactions of the Conference on the Use of Solar Energy, Vol. 2, Part I, p. 74. University of Arizona Press. Kays, W. M. and London, A. L. 1964. Compact Heat Exchangers. New York: McGraw-Hill. Kreider, J. F. and Kreith, F. 1982. Solar Heating and Cooling—Active and Passive Design, 2nd ed. New York: McGraw-Hill. Kreith, F. 1973. Principles of Heat Transfer. Scranton, PA: International Textbook.
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Active Solar Collectors and Their Applications
Kreith, F. and Kreider, J. F. 1978. Principles of Solar Engineering. New York: McGraw-Hill. Kreith, F. and Kreider, J. F. 1980. "Principles of Thermodynamics and Heat Transfer." In Chapter 4, Solar Energy Handbook, Kreider, J. F. and Kreith, F., editors. New York: McGraw-Hill. Lunde, P. 1980. Solar Thermal Engineering. New York: John Wiley & Sons. Thodos, G. 1976. "Predicted Heat Transfer Performance of an Evacuated GlassJacketed CPC Receiver: Counter-current Flow Design." Report ANL-76-67. Argonne, IL: Argonne National Laboratory.
11. SYSTEM MODELS
To err is human, to err by 106 mistakes per second requires a computer. modern folklore
11.1 SYSTEM CONFIGURATION AND CONTROLS
In addition to collectors, a solar thermal energy system contains one or several of the following components: storage, backup, pump(s), heat exchanger(s), pipes, pressure relief valve(s), and controls. Choice of components, system configuration, and control strategy depend on the characteristics of the load. For illustration we show two examples of system schematics in Fig. 11.1.1. Figure 11.1.la shows a closed loop system frequently used for residential applications while Fig. 11. Lib shows a steam system without storage for process heat or power generation. Details will be discussed in later sections. In this section we address only some general aspects of controls which are relevant for most solar thermal systems. The controls are to guarantee that the collectors are turned on if and only if they can deliver useful energy. Obviously the absorbed solar radiation must exceed the heat loss from the collector. According to the instantaneous efficiency equation (4.2.5) [with incidence angle modifier as in Eq. (4.3.1)] the output of a collector field of area A is
with
The collector should be turned on whenever the insolation 7 is above the threshold 7X:
277
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Active Solar Collectors and Their Applications
Fig. 11.1.1 Examples of system diagrams, (a) closed-loop system with storage. (From Klein and Beckman [1979]. Reprinted by permission of John Wiley & Sons), (b) System for process heat without storage.
Which temperature is to be used for 7"con depends on the system. In the system of Fig. 11.1.1 a any heat above the temperature at the bottom of the storage tank is useful. Hence this system should be turned on whenever TcM > Tslot. In the steam system of Fig. 11.1.1 b the outlet temperature must be above a minimum rmin in order to bring about the phase change from water to steam; hence rcol, > rmin. In steam systems the flow rate is adjusted according to insolation to keep TOM constant. In either system the controls require a temperature sensor in the collector. The preferred placement is on the absorber near the outlet. The sensor could also be inside the collector fluid because at the insolation threshold no heat is extracted from the collector and the absorber temperature is close to the fluid temperature. In systems like Fig. 11.1.1 a another temperature sensor must be put at the bottom of the storage tank. In practice one should choose the turn-on and turn-off criteria a little different from the simple values in Eq. (11.1.2) in order to avoid instabilities. The fluid between storage tank and collector inlet is likely to have cooled down while the pump was turned off. If now the pump is turned on when the insolation reaches the threshold Eq. (11.1.2), this cooler fluid may cool the absorber and cause the pump to be shut off again. To prevent pump cycling it is thus advisable to set the turn-on temperature higher than the turn-off temperature; for example, in hot water systems one chooses an increment A Taa on the order of 10°C while the turn-off increment AT^is on the order of 1 °C.
System Models
279
There is another consideration for choosing the increment Aron. The value of the collected heat must be greater than the value of the energy consumed for pumping. If the pump uses electricity at a rate Qpump and if the value of electricity is /?elec [in $/GJ] while the value of delivered heat is />heal, then the output has a net positive value only if
Thus the right-hand side of the turn-on criterion Eq. (11.1.2) must be increased according to
Alcone and Herman [1981] have made the following recommendations for the controller setpoints of solar water heaters, taking into account pumping power, stability criteria, and possible errors in the temperature sensor. Let jTcol| be the temperature read by the sensor in the collector, and assume an error e in this reading. Then the collector pump should be turned off whenever TcM drops below the turn-off value
with
The collector should be turned on when
with
1 \ .2 SYSTEM PERFORMANCE ANALYSIS When designing a solar energy system one needs to understand how the components interact and how the system performs. The system performance depends not only on equipment, system configuration and control strategy but also on climate and location and on the characteristics of the load. Thus the calculation of system performance can be a complicated task. Frequently a computer simulation is deemed necessary, although in special cases shorthand procedures can be used. In fact, much progress has been
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Active Solar Collectors and Their Applications
made in the development of shorthand procedures and by now most of the standard solar energy systems can be designed without a detailed computer simulation. In this section we give a brief survey of the most important methods of solar system analysis. 11.2.1 Computer simulations A computer simulation program is an assembly of the appropriate component models. The components and their interactions are described by algebraic or differential equations. The time resolution depends on the input data. Since weather tapes with hourly data are widely available, the hour has become the most common time step for computer simulations. A simple example will illustrate how a computer simulation works. Consider the domestic hot water system whose schematic is shown in Fig. 11.1. la. Let us neglect heat losses from the pipes and from the storage tank, and let us assume that the tank is always well mixed. The organization of the program is shown by the flow diagram in Fig. 11.2.1. For each hour of
Figure 11.2.1 11.1.la.
Organization of program for simulating the hot water system in Fig.
System Models
281
the year the main program reads the weather data for insolation and ambient temperature. It may also read the hot water demand and the water mains temperature for each hour, but usually a single time-of-day profile of the hot water demand is assumed as standard for every day of the year. On the SOLMET weather tapes [SOLMET, 1978] both the beam normal insolation and the hemispherical insolation on the horizontal surface are given1 as hourly irradiation values [in kJ/m2]. The corresponding average irradiance [kW/m2] during the hour is obtained by dividing by 3600 sec. When the beam normal irradiance 7b and the hemispherical horizontal irradiance 7h are known, the diffuse irradiance 7d is
where 0Z is the zenith angle of the sun. Now the program calls the insolation subroutine to calculate the irradiance on the collector aperture (taken as flat plate) according to
where 6 is the incidence angle on the collector, /? is the collector tilt, and Pground is the reflectance of the ground (see Section 3.2). Next the collector subroutine evaluates Eq. (11.1.1) with F = FxFin and rcoll = r HX ,m because the inlet temperature to the heat exchanger is known (and equal to the storage temperature). Multiplication by A£ = 3600 sec yields the energy collected during the hour as
with
The plus sign indicates that only positive contributions are to be counted. Finally the storage subroutine calculates the new storage temperature
The last term represents energy withdrawn from the storage tank by the load. The load data are given in terms of the water flow Amioad consumed by the load during each hour. This water is withdrawn at the temperature T"stor 'The beam "data" of the SOLMET tapes are actually calculated as discussed in Section 3.4. If the beam insolation is not given, it must be estimated from the hemispherical horizontal insolation by one of the algorithms in Section 3.5.
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Active Solar Collectors and Their Applications
and replaced by water from the mains. The backup is in series with the solar system and boosts the temperature to the specified use temperature rload if necessary. If rstor is above r,oad, Eq. (11.2.4) assumes that the user tempers the hot water with cold water, thus withdrawing no more energy than needed; i.e., (Amc)ioad(rload - Tmains). The resulting storage temperature !Ts,or; new is brought to the main program as input for the next hour. The main program keeps track of the energy supplied by solar and by backup (also called auxiliary) by summing
and
The following example illustrates this procedure. EXAMPLE 11.2.1
A parabolic trough is to be used for a process steam application. The operating temperature range is 180-270°C, and the corresponding linearized efficiency curve can be characterized by intercept jpm?70 = 0.65 and slope FmU -0.5 W/m2 °C. The incidence angle modifier is given by
The system configuration is as shown in Fig. 11.1.1 a but without heat exchanger between collector and storage tank. The fluid in collector and tank is oil with c = 2 kJ/kg °C, and the tank is well mixed. The collector flow rate is constant at m = 0.5 kg/sec and the aperture area is A = 100 m2. The storage mass is M - 5000 kg and the load is only at night. Calculate the storage tank temperature and energy input to storage as a function of time for a sunny equinox day assuming the collector is deployed with horizontal east-west tracking axis, at a location at 35° latitude and sea level. (Use Hottel's clear day model of Section 3.5.2 and assume constant ambient ramb = 15°C.) The storage tank starts out at 180°C in the morning. SOLUTION
First one has to convert the efficiency parameters from mean fluid temperature Tm to fluid inlet temperature T-m. Equation (4.2.9) yields
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Then
The irradiance on the aperture / = 7b cos 0 is listed in the table below: Time(h) 6
5 -44 -33 -22 -1 0 1 2 3 4 5 6
Irradiance on aperture (W/m2)
Collected heat (kJ/m2)
Storage temperature (°C)
0 0 223 223 776 776 1194 1194
180.0 180.0 182.2 190.0 190.0 201.9
1439 1520 1387 1096 646 78 0 0
216.3 231.5 245.4 256.4 262.8 263.6 263.6 263.6
0
88 288 288 498 498 671 671 782 821 782 671 498 288 88 0
The collected heat for each hour and the resulting new storage temperature are then calculated according to Eqs. (11.2.3) and (11.2.4). At the start of the day, rstor = 180°C and the heat loss term
this is larger than the absorbed solar radiation between 6:30 and 7:30 am (equals hour "-5"):
Hour " — 4" (7:30 to 8:30 am) is the first hour with positive collector output:
This raises the storage temperature by 22.3 MJ/(5000 kg X 2 kJ/kg °C) = 2.2°C.
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Active Solar Collectors and Their Applications
The results are also listed in the table. For a complete computer simulation this calculation is repeated for each day of the year, using real weather data. This has been a simple program for a simple and somewhat idealized system. Any amount of detail can be included in a computer simulation. In the above example one could easily include additional features. Heat losses from the storage tank are treated by adding a term — (UA)aor(TMr — T1env) to the right-hand side of Eq. (11.2.5), (7slor and ^4stor being the conductance [in W/m2 K] and surface area of the storage tank and Tenv the temperature of the tank environment. Heat losses from pipes can be included most conveniently by modifying the collector parameters ija and U according to Section 10.5. If losses due to transients are expected to be significant, then the warmup and cool-down of the collector field should also be modeled [see Eqs. (4.5.3) through (4.5.6)]. The severity of losses due to transients depends very much on the characteristics of a specific application. In systems such as those of Section 12.3.2 and 12.3.3 losses due to transients are usually negligible because the collector acts as preheater. In fact if the inlet temperature is equal to ambient, the transient losses are 0 because any heat used for warming up the collector in the morning will be extracted in the evening. In some high-temperature systems on the other hand, transients can be important if heat below a certain threshold cannot be utilized. Some programs use a refined method for treating differential equations that govern components such as storage. For example, Eq. (11.2.4) is really a finite difference approximation to the differential equation
with time step At = 1 h. Equation (11.2.4) bases the collected energy A(2 and the energy delivered to the load on the storage temperature at the beginning of the hour. However, in actuality rslor changes during the hour, and A(2 and load should be based on the average storage temperature during the hour. A simple equation like Eq. (11.2.7) can be solved in closed form [Lunde, 1980]. In more complicated situations, such as partially stratified storage tanks, one will resort to numerical solutions. There are several possible algorithms for solving such differential equations, for instance, the Euler method and the Runge-Kutta method. The TRNSYS [Klein et al., 1979] simulation program uses the following modified Euler method. Suppose the dependent variable T [e.g., the storage temperature in Eq. (11.2.7)] is given by a differential equation of the form
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If T has been found to have the value T0 during the previous time step then the first prediction for the new value is
Now Tp can be used to calculate a new value of the derivative f(t, Tp, other variables). Using the arithmetic mean of the old and the new derivative one obtains a corrected value Tc:
The process can be repeated using Tc as a new value for Tp until Tf and Tc differ by less than a specified error. The storage equations are more complicated if the storage tank is stratified with the warmest fluid at the top and the coldest at the bottom. In this case one can divide the tank into a number of horizontal sections or nodes and assume that the incoming fluid flows to the node that best matches its own density. High stratification is desirable because it reduces the collector inlet temperature. In practice most storage tanks fall somewhere between the extremes of perfect mixing and perfect stratification. Thus it appears [Duffie and Beckman, 1980] that a rather small number of nodes (say, three nodes) can provide a very good approximation. In order to give a meaningful representation of system behavior, the input weather data should be typical. In particular, one should not use an unusually sunny or an unusually cloudy year. One could run the simulation with data for many years but that would increase the cost unnecessarily. As a better alternative one can synthesize a typical meteorological year2 by splicing together those months from the long term record that best represent the long term average. Several simulation programs for active solar energy systems are available for public use. Examples are TRNSYS [Klein et al., 1979] and SOLTES [Fewell and Grandjean, 1979]. They are powerful and flexible, and contain a large number of subroutines for each of the many components that could be used in a solar energy system. Their run times depend on the complexity 2 The SOLMET data tapes contain what is called typical meteorological year data ( = TMY). They were extracted from the long term record (about 20 years at most stations) by the following procedure: First the long term average monthly values of insolation, temperature, humidity, and windspeed were calculated. Then those months were selected from the long term record that agreed most closely with the long term averages. The selection criterion assigned different weights to insolation, temperature, etc., according to their importance in most solar calculations [Hall et al., 1978].
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Active Solar Collectors and Their Applications
of the system and on the level of detail to be modeled; typical run times for a 1-year simulation range from minutes to 10s of minutes on a big computer like the IBM 360. Computer simulations can be a valuable tool, but they are not without problems. Above all, computers do not think. Judgment is needed both in the selection of the input and in the evaluation of the output. One should be critical of the false sense of confidence that may result from the sheer quantity of detail included in the program. Second order details may be meaningless when first order effects remain uncertain. For example, uncertainties in insolation, load, and economic variables are usually quite large. The very flexibility of the big simulation programs has its drawbacks. So many variables must be specified by the user that there is a risk of errors in interpretation or specification of the input.3 The user does not even know beforehand which variables will be important and which will not. The instruction manuals alone are hundreds of pages thick, and a fair amount of preparation is required before one can use the programs. Such expertise can be costly. In addition there are expenses for running the program. While a few runs are quite affordable, the cost of a system optimization with these programs may well become excessive because a numerical optimization requires evaluation of a very large number of cases. Suppose, to take a simple example, one wants to optimize a domestic hot water system with heat exchanger. For each system schematic and for each selection of equipment manufacturers there are several design variables to be optimized: collector area, heat exchanger area, flow rate in collector loop, flow rate in storage loop, and storage capacity. If one considers n possible values for each of these variables, the total number of cases is «5. Five values per variable does not give a fine resolution, yet it requires 5s = 3125 simulations for just one system schematic. If one is confident that the functional dependences are sufficiently smooth, one can abbreviate this procedure considerably by using clever search procedures (e.g., evaluating gradients). Nonetheless, the problems of optimization remain so severe that in practice one does not use simulation, programs like TRNSYS directly for system optimizations [Bruno, 1979]. 11.2.2 Shorthand procedures As an alternative to computer simulations, several shorthand procedures (also called design tools) have been developed. They are suitable for certain standardized solar energy systems, and they are simple enough for programmable calculators or even hand calculation. Best known among these procedures are the/chart for active space heating, and the utilizibility method. Some of these procedures are quite fundamental and general; they are 3
This has been observed in attempts to validate simulation programs for energy consumption of buildings. While the programs appear to be more or less correct, input errors are common [Kusuda, 1980].
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described in this chapter. Others are limited to specific applications and will be addressed in Chapter 12. Shorthand procedures can be developed by several techniques, singly or in combination: curve fitting of computer simulation results, statistical analysis of weather data, and analytical solution of the system equations.4 Whether one should choose a shorthand procedure or a simulation program depends on the task at hand. A simulation program is a research tool for exploring systems that are not well understood or that are too complicated to be described by shorthand procedures. Simulations are essential for validating shorthand procedures. Simulations may also be necessary if one wants to investigate parameter ranges, operating conditions, or design details that are not covered by the available shorthand procedures. By now shorthand procedures have been developed for many if not most of the standard solar thermal applications. There is no need to employ computer simulations for the design of a standard residential solar system or the design of hot water systems for industrial processes with uniform load. On the other hand, for the design of a solar cogeneration5 system one may need a computer simulation. With shorthand procedures the input is simple, and so is the output. The calculations are fast enough to make system optimization feasible. In some cases one can even determine the optimum analytically in closed form. As for accuracy, good simulation programs have errors below 1% (assuming that the input is exact). Shorthand procedures have errors in the range of 2%-5%, quite acceptable under most conditions since the input has much greater uncertainties. 11.3 YEARLY COLLECTIBLE ENERGY The simplest shorthand procedures involve correlations for yearly collectible energy. At the first and easiest level they predict the yearly total radiation incident on the aperture of a collector. These correlations have already been presented in Section 3.8; they are useful for technology assessment, as well as for the analysis of systems whose efficiency is nearly independent of insolation level (e.g., photovoltaics). At the second level, described in Section 11.3.1, the correlations include a threshold to account for heat loss; thus they can be used for the analysis of industrial process heat systems and solar thermal power plants. The correlations have been developed for the following collector types [Rabl 1981]: flat plate and evacuated tubes: Fixed collector with tilt equal to latitude, facing due south "When correlating simulation results, one is well advised to first develop an understanding of the underlying principles; otherwise one may end up fitting noise rather than meaningful trends. 5 A cogeneration system (also called total energy system) supplies several energy forms at the same time; e.g., a photovoltaic cogeneration system would provide not only electricity but useful heat, extracted by cooling the cells.
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Active Solar Collectors and Their Applications
CPC: Compound parabolic concentrator with geometric concentration ratio C = 1.5, acceptance half-angle 35°, with fixed tilt equal to latitude, and facing due south east-west: Collector with aperture tracking about horizontal east-west axis north-south: Collector with aperture tracking about horizontal northsouth axis polar: Collector with aperture tracking about polar axis (i.e., about north-south axis with tilt equal to latitude). 2-axis: Collector with 2-axis tracking whose aperture is always normal to sun (e.g., parabolic dish and point-focus Fresnel lens) tower: Power tower or central receiver of a design characterized by the optical efficiency-incidence angle relationship in Table 7.2.1b The only climatic input6 needed is the yearly average daytime beam normal irradiance 7,, of Fig. 3.4.3. Despite their simplicity these correlations are quite accurate: they reproduce the results of hour-by-hour simulations with an accuracy (rms error) of 2% for flat plates and 2%-4% for concentrators. Certain standardizing assumptions had to be made for the development of this procedure; they are summarized in Table 11.3.1 and discussed in Section 11.3.2. To illustrate the use of the yearly correlations let us consider photovoltaic systems. The conversion efficiency of a photovoltaic cell is nearly independent of insolation. Once the yearly average temperature of the cell has been determined for a given cell design and cell cooling mode, one can approximate the corresponding cell efficiency by a constant ??e (= solar-to-electric 6 This remarkable fact can be summed up by the statement: "to a solar collector all climates look very much alike and differ only in their average beam normal insolation."
TABLE 11.3.1 Assumptions Used for Deriving Correlations for Yearly Collectible Energy3 (i)
Collector uses all solar radiation above a specified threshold X, and the portion of the solar radiation above X is used with constant efficiency, (ii) Hourly time resolution, with neglect of transients, (iii) Flat plate accepts beam and diffuse insolation, and diffuse component is isotropic. CPC accepts beam and — diffuse. Tracking collectors accept only beam insolation, (iv) No shading for flat plate and CPC. For tracking collectors (except tower), standard collector spacing with ground cover ratio i/- = 0.5 for collectors tracking about one axis, and $ - 0.25 for collectors tracking about 2-axis. (v) Incidence angle modifier of Eq. (11.3.8) for flat plate, CPC, and 1-axis trackers. (vi) For tower incidence angle modifier K(0,, s) of Table 7.2.1 b, but corrected by a factor I/ 0.90 to remove the reflectivity assumed in this table. "From Rabl [1981].
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289
efficiency). The yearly electricity production is then obtained by multiplying the yearly incident irradiation q0 for the collector type in question by %• EXAMPLE 11.3.1
Consider a flat plate photovoltaic panel at a tilt angle equal to the latitude and with cell efficiency 77,, = 10%. From the contour map in Fig. 3.4.3 one sees that the sunniest region of the United States has insolation values around 7b = 0.6 kW/m2. Reading the solid line in Fig. 3.8.1 corresponding to X = 35° (Albuquerque), one finds an available solar irradiation of q0 = 8.0 GJ/m2. The corresponding electricity production is
The lowest beam irradiance occurs in the northwest and northeast. Reading the solid line corresponding to X = 45° and 7b = 0.23 kW/m2 (northern part of New York State), one finds an electricity production of 0.42 GJ/m2 for the same cell efficiency. EXAMPLE 11.3.2
Consider a parabolic dish with rim angle 40°, geometric concentration ratio 1000, and total optical error characterized by an rms beam spread of 7 mrad. Assume a high-efficiency cell with TJC = 25%, and a reflectance of the reflector of p = 0.9. Let us include the effect of circumsolar radiation, since doubts have frequently been raised about the loss of performance of focusing collectors in regions with high circumsolar radiation. The correlations for the available irradiation q0 for focusing collectors are based on pyrheliometer data and require correction factors if the acceptance half-angle of the collector is smaller than the 2.6° acceptance half-angle of the pyrheliometer. This correction factor is given by the intercept factor, Eq. (8.6.2),
where Rm(X) is the long term average circumsolar ratio for the location in question, and -ys, av and yc, av are the intercept factors for the solar disk and for the circumsolar region. Numerical values for 7^ av and yCj av are listed in Table E.4 of Appendix E for typical parabolic dish designs; for the present example, they are and
290
Active Solar Collectors and Their Applications
The monthly average circumsolar ratios R^(X) are listed in Table E. 1 as a function of threshold X for all locations for which data are currently available. For photovoltaic applications, the appropriate threshold is X = 0. Assuming that a simple average over all of the months listed in Table E. 1 is a fair indication of the long term yearly average, we find a value Rav( X = 0) = 0.040 for Albuquerque, a location with very low circumsolar radiation. This yields an intercept factor of 7av = 0.960. Of the insolation q0 — 9.2 GJ/ m 2 (Fig. 3.8.1) available to a two-axis tracking concentrator, a fraction p%v is reflected by the reflector and intercepted by the cell, and hence the electricity production is
if p = 0.90 and r,e = 0.25. Of the locations for which circumsolar radiation measurements have been made, Argonne (near Chicago, IL) has the highest circumsolar ratio, 0.108 averaged over the months in Table E. 1 . The corresponding intercept factor is 0.936, somewhat smaller than for Albuquerque. The available insolation
11.3.1 Radiation above a specified threshold Thermal collectors are turned on only when the solar radiation on the aperture exceeds the heat loss threshold. If all days and hours were identical, the delivered energy could be obtained by simply subtracting the total daily heat loss from the radiation absorbed by the collector. Due to the variability of the weather, the true energy gain can be significantly higher. This feature can be illustrated by the following two artificial climates. Climate 1 has identical days, all uniformly overcast, while climate 2 has clear days half of the time and no sunshine for the rest; both climates have the same long term average insolation H. If the heat loss of a collector equals the peak insolation of climate 1, no useful energy can be collected. Under climate 2, however, the same collector can collect some useful energy on the clear days. To account for this effect it is convenient to define function , called utilizability, as ratio of the radiation above a threshold X and the total radiation:
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291
Monthly expressions for 4> are given in Section 11.4. In the present section <{> is not used explicitly but it could be calculated, if desired, as
To include the effect of a threshold, the long term average yearly energy Q per aperture area A has been correlated in the form Q/(AFrj0) with a polynomial
of second degree in the threshold X
where q0, q\, and q2 are polynomials of first or second degree in latitude and average direct normal insolation. Note that in these correlations the threshold is stated in terms of the optical efficiency at normal incidence even though the result for q(X) actually includes the incidence angle modifier [i.e., the difference between X and 7X of Eq. (11.1.2)]. The heat transfer factor Fin the instantaneous efficiency equation cancels when the threshold is calculated and hence the same correlation can be used regardless of temperature base (i.e., Tabs, Tm, Tm, or THX.HI)- These correlations are plotted in Figs. 11.3.1-11.3.7 and the equation for the curve fit is included under each figure. The correct units must be used for 7b, X, and L in these curve fits: 7b = yearly average beam normal irradiance during daylight hours, in kW/m2 X = ratio of heat loss and optical efficiency, in kW/m2 L = geographic latitude, in radians q(X) has units of GJ/m2. (These units were chosen in order to avoid very large or very small coefficients in the correlations.) For polar, east-west, and 2-axis the latitude dependence is negligible; for the other collectors the latitude has been included as a variable in the fits. Correlations for several evacuated tubes have been published by Rabl [1981]; they correspond to the incidence angle modifiers in Fig. 4.3.2. Despite drastic differences between the incidence angle modifiers, the resulting correlations are remarkably close to each other. The Owens Illinois curves are almost identical to those for the flat plate, and the others show differences on the order of ± 5%. The precise shape of the incidence angle
292
Active Solar Collectors and Their Applications
Fig. 11.3.1 Yearly collectible energy for flat plate at tilt equal latitude; shown as graph and as curve fit for Q/(AFij0), as function of threshold X (in kW/m2), of average direct normal irradiance 7b [in kW/m2], and of latitude (X in degrees, L in radians) (From Gordon and Rabl [1982]).
modifiers does not appear to be critical for the calculation of yearly average performance. In fact, one obtains essentially the exact result if one simply rescales the flat plate correlation, Fig. 11.3.1, by the appropriate ratio of the all-day average incidence angle modifiers. The incidence angle in Fig. 4.3.2 is in the direction perpendicular to the tubes. For a collector at tilt equal latitude with tubes running north-south, the projected incidence angle for Fig. 4.3.2 is therefore the hour angle; hence the correction factor is
The number 0.92 is the average of the reference incidence angle modifier, Eq. (11.3.8). With this correction one can use Figs. 11.3.1-11.3.7 even for collectors whose incidence angle modifier is very different from Eq. (11.3.8).
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Figure 1 1.3.2 Yearly collectible energy for CPC, shown as graph and as curve fit for Q/(AFri0), as function of threshold X (in kW/m2), of average direct normal irradiance /,, (in kW/m2), and of latitude (X in degrees, L in radians) (From Gordon and Rabl[ 1982]).
EXAMPLE 11.3.3 To see how the correlations can be used for thermal collectors, consider a parabolic trough for which the following parameters have been measured:
This collector is to operate in Albuquerque, NM, at an average fluid temperature Tm = 300°C. The yearly average ambient is Tamb = 13°C and the average daytime beam irradiance is 7b = 0.60 kW/m2 (from Fig. 3.4.3). SOLUTION The threshold is
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Active Solar Collectors and Their Applications
Figure 11.3.3 Yearly collectible energy for concentrator tracking about east-west axis, shown as graph and as curve fit for Q/(AFri0) as function of threshold X (in kW/m2), and of average direct normal irradiance lh (in kW/m2) (From Gordon and Rabl[1982]).
Interpolation between the Ib = 0.55 and 0.65 curves in Fig. 11.3.3 yields q(x) - 3.65 GJ/m2. Hence this collector can deliver Q/A = Fm-r,0q(x) = 2.37 GJ/m2 per year if it is installed with east-west tracking axis. To illustrate the effect of dirt and degradation, let us suppose that the optical efficiency is reduced by 10% to Fmri0 = 0.585, averaged over the lifetime of the collector, while the U value remains constant. Then the threshold increases to 0.327 kW/m2 and the yearly output is only 1.96 GJ/m2. A second example will illustrate the usefulness of these correlations for industrial process heat and for solar thermal power plants. EXAMPLE 11.3.4
Suppose water from the water mains is to be heated to 90°C, e.g., for cleanup operations in the food industry. The inlet temperature Tm of the collector-is
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Figure 11.3.4 Yearly collectible energy for concentrator tracking about horizontal north-south axis, shown as graph and as curve fit for Q/(AFria) as function of threshold X (in kW/m2), of average direct normal irradiance ~Ib (in kW/m2), and of latitude (A in degrees, L in radians) (From Gordon and Rabl [1982]).
nearly constant, either at the yearly average ambient ramb or at a constant amount above ramb if a heat recovery unit is installed as a conservation measure. Flow rate m and collector area A are chosen so that an outlet temperature Tout = 90°C is reached only under peak insolation, which typically means
(This is the single-pass open loop design discussed in Section 12.3.) Combining this condition; i.e.,
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Active Solar Collectors and Their Applications
Figure 11.3.5 Yearly collectible energy for concentrator tracking about polar axis, shown as graph and as curve fit for Q/(AFr]0),_as function of threshold X (in kW/ nr), and of average direct normal irradiance 7,, (in kW/m2) (From Gordon and Rabl[1982]).
with Eqs. (4.2.4) to (4.2.6) for 77 and F-m yields the flow rate-heat capacitance product me as
Suppose the collector is a Fresnel lens plus mirror collector in polar mount with the parameters Fmij0 = 0.656, and FmU = 0.78 W/m2 °C as reported in Collares-Pereira [1979]. Let us assume operation with T-m = Tamb; then the threshold is X = 0, and one obtains from Eqs. (11.3.6) and (4.2.4) the value
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Figure 11.3.6 Yearly collectible energy for concentrator with 2-axis tracking, shown as graph and as curve fit for Q/(AFr]0), as function of threshold X (in kW/ m2) (From Gordon and Rabl [1982]).
From_Fig. 11.3.5 for the polar case, we read q(X) = 8.15 GJ/m2 for locations with 7b = 0.6 kW/m2; thus the yearly collectible energy per aperture area is
For a direct comparison between several typical collectors, we plot in Fig. 11.3.8 the yearly average energy delivery Q/A versus operating temperature (as AT = Tm — ramb = difference between fluid and ambient temperatures) at a geographical latitude of 35°. Two climates are shown: (a) a sunny climate with_average beam irradiance 7b = 0.6 kW/m2, and (b) a fairly cloudy one with /„ = 0.3 kW/m2. The collector parameters are listed in Table 11.3.2; they are typical values based on standard outdoor tests. One of the most striking conclusions from this comparison concerns the crossover
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Active Solar Collectors and Their Applications
Figure 11.3.7 Yearly collectible energy for central receiver, shown as graph and as curve fit for Q/(AFpoi), as a function of threshold X (in kW/m2), and of average direct normal irradiance 7b (in kW/m 2 ). This correlation is based on the particular design with incidence angle modifier of Table 7.2. Ib (195 MWft plant at 35° latitude). For the incidence angle modifier of Table 7.2. la the collectible energy would be significantly higher (From Rabl [1981]).
between flat plate and concentrating collectors. At temperatures that are more than approximately 25°C above ambient, all of the concentrating collectors in Fig. 11.3.8 surpass the flat plate in performance. This conclusion holds even for cloudy climates where concentrating collectors were believed to be at a disadvantage because they miss most or all of the diffuse insolation. The high heat loss of flat plate collectors turns out to be more important: at times when the insolation is above the threshold for a flat plate collector (with AT > 25°C) there is enough direct radiation to run a concentrating collector. Of course, energy delivery is only the denominator of the cost per energy criterion; the choice of a collector will depend just as much on its cost.
Figure 11.3.8 Yearly collectible energy vs. average difference between fluid temperature and ambient for typical collectors: (a) 7b = 0.6 kW/m2 and (b) /„ = 0.3 kW/m 2 (From Rabl [1981]).
299
300
Active Solar Collectors and Their Applications TABLE 11.3.2 Typical Collector Parameters Assumed for Fig. 11.3.8 Collector type Flat plate, double glazed" CPC 1.5^ with evacuated tubes Parabolic troughd Parabolic dishd Power towerf
^V/o
FmU [W/m 2 °C]
0.70b 0.60C
0.75C
0.65e 0.61C pa = 0.818
0.67 0.27 0.16
5.0
"Average of four best double-glazed collectors, Florida Solar Energy Center, Summary Test Package, issued 15 February 1978, and Supplements 1 (April 1978) and 2 (June 1978). b With respect to pyranometer. "With respect to radiation within acceptance angle. d Typical test results from Dudley and Workhoven [1979] e With respect to pyrheliometer. 'Projected performance. g p = heliostat reflectance; a = receiver absorptance.
11.3.2 Assumptions for yearly correlations To explain the assumptions we give a brief description of the computer program that was used to derive and validate the correlations; for a more complete discussion the reader is referred to Rabl [1981]. For each hour of the year, the beam irradiance 7b (at normal incidence) and the hemispherical irradiance /h (on horizontal surface) are read from the SOLMET weather tapes. The insolation values are interpreted as constant values during that hour, and all incidence angles are calculated for the midpoint of each hour. The diffuse irradiance
with 0j = incidence angle on the ground, is assumed to be isotropic. The irradiance Icoll incident on the collector aperture is calculated by including the appropriate incidence angle modifier K(0) for the optical efficiency. i;0 is the optical efficiency at normal incidence, and rj0K(6) is the optical efficiency when the incidence angle is 0. Since incidence angle modifiers vary as much between collectors of the same genetic type as between different collector types, it appears reasonable to select one typical modifier and use it for all collector types other than the power tower. For this purpose, the incidence angle modifier of the SERI Hexcel collector of Gaul and Rabl [1980] was chosen; it is
with 6 in degrees. The average of K(6) for isotropic radiation is
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The optical efficiency ?j0 is left out of the simulation and applied as a multiplicative factor to the end result. Thus the irradiance / on the flat plate is
where 0 = incidence angle on aperture A = tilt = latitude Pground = reflectance of ground, assumed to be 0.2 without, and 0.7 with, snow cover For a CPC with geometric concentration ratio C, the insolation on the aperture is taken as
when the beam component is within the acceptance angle, and 0 when it is outside. The tracking collectors are assumed to accept only the beam component of the insolation. For the central receiver, the incidence angle modifier depends on zenith and azimuth angles and is obtained by interpolating the numbers in Table 7.2.1b. For consistency with the numbers in Table 7.2.1b the aperture area of the central receiver is denned as the surface area of the heliostats. In most installations, several collector modules will be connected to each other, and hence they may shade each other during some periods. Since shading is installation dependent, we have made certain assumptions about the deployment of collector modules. For flat plates, we have neglected shading altogether, since flat plates are frequently installed on roofs of houses where they do not interfere with each other. For CPCs we have also neglected shading because the limited acceptance angle rules out shading during operating hours, at least for reasonable collector spacing at low and intermediate latitudes. For collectors with 1-axis tracking, the analysis of shading is straightforward if, as will usually be the case, the collector rows are long enough to permit neglect of end effects. Let $ be the ground cover ratio; i.e., the ratio
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Active Solar Collectors and Their Applications
of the aperture area A of the collector array and of the ground area A% that is taken up by the collector array. When the collector rows do not shade each other, the beam irradiance on the collector array is
Whenever there is shading, no beam insolation reaches the ground, and hence the irradiance available to the collector array is
where 0Z is the incidence angle of the sun on the horizontal surface. Hence the irradiance per aperture area of 1-axis tracking collectors is
For arrays of collectors with polar tracking axis this formula is correct if ^ is denned in the plane of the tracking axis. We assume this to be the case, since such collectors are likely to find their main application in relatively small installations with a single plane of collector modules; e.g., on a tilted roof. For simplicity, we have used Eq. (11.3.13) even for the 2-axis case, where it provides an upper bound rather than the exact result for the available insolation. We have taken a ground cover ratio
as a typical value for all collectors with one-axis tracking. Results for other ground cover ratios can be found in Kutscher et al. [1982]. For the 2-axis case we assumed the value7
For the central receiver shading and blocking effects are included in the incidence angle modifier of Table 7.2.1. To account for heat losses, a threshold X is defined as ratio of heat loss and optical efficiency. A range of threshold values (X = 0, 0.1, 0.2 .. . 0.9, 1.0 kW/m2) is considered, and for each value of X the collectible insolation (7co,i — X) is summed to obtain the yearly total
'Because of the different ground cover ratios, the ratio of radiation availability for 2-axis and for polar differs from the yearly average, 0.96, of the cosine of the solar declination.
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for each of the seven collector types, the + subscript on the bracket indicating that only positive summands are included. Transient effects are neglected; they could be included, using the formulas of Zarmi [1982]. Having calculated the yearly collectible energy q&M(X) as a function of threshold X for each of the 26 SOLMET stations, we fitted the results with polynomials in threshold, latitude, and average beam irradiance. Our criterion for the best fit is the minimization of the sum of squared differences between data and fit:
the sum running over all 26 stations and over the threshold values X = 0 to 0.9 kW/m2. This requires the solution of a system of n linear equations
for the n coefficients a, in the curve fit. Having derived such fits, one would like to determine how much discrepancy is to be expected between the fits and the exact answer, the latter being defined as the result of the hour-by-hour simulation. Two measures of the discrepancy are of interest: the mean error or bias, and the rms error or standard deviation. The bias is defined as
in relative units. The relative rms errors are defined as
The rms errors are typically around 2% for flat plate and CPC, and around 2%-4% for tracking collectors. The bias is a small fraction of a percent. Our results for yearly collectible energy agree with other investigations8 to the extent to which the same data and assumptions were used. There are some differences, mainly because the direct insolation "data" used in previous investigations were based on an earlier model, and the previous investigations did not include incidence angle modifiers and shading effects. The simulation described above assumes a constant threshold. In a real system the threshold may vary both with ambient and with collector tem8
For example, Boes et al. [1977] and Dickinson [1978].
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Active Solar Collectors and Their Applications
perature. To evaluate the effect of threshold variations, we have repeated the simulation with thresholds that vary about their annual average in typical diurnal and seasonal patterns:
with n the day of year and t the time of day [hours]. The ratio of maximum to minimum is quite large, 3:1. The results indicate that threshold variations have only a very small effect for thresholds below 0.2 kW/m2. The principal difficulty of this analysis lies in choosing the variables and the number of coefficients for the polynomial fits. With a finite data base, one does not necessarily improve a fit by increasing the number of coefficients; instead, one may be fitting spurious fluctuations. This problem of fitting the noise rather than the real trend is particularly serious if the fit is nonlinear and if the data base is not uniformly distributed over the range of variables that one is trying to fit. The fits in this section seem to be the best compromise if one chooses a polynomial. There are, however, other functional forms that could yield better fits. By carefully examining the underlying principles Gordon and Zarmi [1982a and 1982b] have derived closed analytical expressions for q. In particular they calculated the behavior of q at small and large thresholds (i.e., X -* 0 and X -*• 7max) where 7max is the peak solar irradiance on the aperture during any time of the year. /raax depends on climate and on collector type and is the value of X at which q(X) vanishes. 7max is approximately 0.85-1.10 k W/m for flat plates (cloudy to clear climates, respectively) 0.80-1.00 k W/m for concentrators (cloudy to clear climates, respectively).
At 7max the yearly total energy must vanish at least quadratically:
305
System Models Thus it turns out that the functional form
would be a better fit. The exponent a depends on climate. Gordon and Zarmi find that
gives a good fit for focusing collectors where
is the average irradiance on the collector aperture during daylight hours. 11.3,3 Operating time The pumping energy can impose a significant penalty on active solar thermal systems. In order to calculate the pumping energy one needs to know how long the pumps are running. The operating time of the collector loop can be obtained directly from the utilizability correlations [Mitchell et al., 1981; Gordon and Rabl, 1982]. For this purpose it is convenient to plot the time distribution of insolation values as cumulative time distribution. The cumulative distribution is obtained by ordering the instantaneous insolation values according to their magnitude. Such a distribution is sketched in Fig. 11.3.9. The x axis is the cumulative time r, and the y axis the insolation I(r) on the aperture. The curve / on the aperture is defined in such a way
Figure 11.3.9 Schematic sketch of cumulative time distribution I(r) of irradiance on collector aperture, and operating time for threshold X.
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Active Solar Collectors and Their Applications
that the irradiance 7 on the aperture is less than /(T) for T seconds of the year / < /(T) for T sec of the daylight portion of the year.
(11.3.28)
Only daylight hours are included in Fig. 11.3.9; otherwise the graph would have to be extended to the left with 1.58 X 107 sec of zero insolation, not very informative. The total area under the curve equals q0, the yearly irradiation incident on the aperture. The area above the threshold X and below the curve equals q(X), the yearly irradiation above X,
where rmax = length of daylight portion of year = % year « 1.58 X 107 sec and rx is the value where the integrand vanishes
The yearly total operating time rop of the collector is the difference between TV and T ma v
and it is readily found by taking the derivative of Eq. (11.3.29). On the righthand side the only X dependent terms are X and TX; hence the derivative is
The first term vanishes by Eq. (11.3.30) and the second term integrates to the operating period. Thus one obtains the fundamental result that the yearly total operating period is minus one times the derivative of the collectible energy q(X) with regard to threshold:
This formula is exact if q(X) is known exactly. Even though the quadratic correlation reproduces the collectible energy within 2%-4%, its derivative is far less reliable. Thus the formula
turns out to have errors on the order of 10%. Fortunately that is acceptable for most applications. The astute reader may notice why a second order
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307
polynomial is an awkward choice for q(X). The limits of Eq. (11.3.26) fix both g, and
if one uses the numerical values of #, and q2 from the correlations in Figs. 11.3.1-11.3.7, with Xin kW/m2. This is illustrated in the following example. EXAMPLE 11.3.5
A flat plate collector is deployed at tilt 0 equal latitude X at a location with X = 32°N and yearly average temperature ramb = 10°C and average daytime beam irradiance 7b = 0.532 kW/m2. The collector efficiency parameters are Fitflo = 0-75 and FinU = 6 W/m2 °C. The inlet temperature is constant at Tin = 40°C. Calculate the daily average operating time. SOLUTION
The correlation of Fig. 11.3.1 yields for this case q(X) = 7.384 - 13.40JST + 6.Q46X2; hence the numerical values of q\ and q2 are q\ = 13.40 and q2 = 6.046. The threshold is
The daily average operating time is, from Eq. (11.3.34),
11.3.4 Other thresholds The heat loss is not the only threshold that may be of interest. The correlations in this section can be used for the analysis of several other threshold effects. If a system has a constant load and if the peak output of the system
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Active Solar Collectors and Their Applications
exceeds that load then the amount of delivered energy above that load can be calculated. Consider, for example, an industrial process with constant load L [in kW] served by a collector field of area A, with efficiency parameters »;0 and U, operating at AT above ambient. The heat loss threshold is
If the field is large enough, the instantaneous output will be greater than the load
whenever the insolation / is above the second threshold
From the correlations one readily calculates Q(X2), the energy that must be either put into storage or discarded. The same idea can be used for photovoltaic systems, as the following example shows. EXAMPLE 11.3.6
A factory in southern Florida (X = 25°, 7b = 0.35 kW/m2) has a constant electric load of L = 100 kW, and one can place 50 m X 50 m of flat photovoltaic panels at tilt equal latitude on its roof. The utility has agreed to buy the excess electricity production at 5
The threshold for sellback is
For photovoltaic systems Fri0 is interpreted as electric conversion efficiency r)c and hence
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309
can be sold per year at a price of $6500. The total electricity production is
1 1 .4 MONTHLY UTILIZABILITY METHOD For some solar energy systems a simple annual analysis is not appropriate. More detailed calculations are necessary for systems that operate only part of the year (e.g., space heating), for systems that discard excess energy during part of the year, and for equipment whose efficiency varies with time of year. The month is a convenient interval, fine enough to discern seasonal trends yet large enough to permit analysis by hand calculator. In this section we present the monthly utilizability method. The utilizability concept evolved in the classic work of Hottel and Whillier [1958] and of Liu and Jordan [1963]. It has recently been simplified and generalized to include both flat plates and concentrators [Collares-Pereira and Rabl, 1979a, 1979b]. An equivalent formulation, but for flat plates only, has been developed by Klein [1978]. The monthly utilizability method serves as basis for many calculations, in particular the 0, /chart of Klein and Beckman [1979] for the analysis of certain systems with storage. The calculation proceeds in two steps. First one calculates the monthly average daily total solar radiation H on the aperture. This is sufficient for systems whose efficiency is independent of insolation level, e.g., photovoltaic systems. For systems with a threshold (e.g., thermal systems), one needs a second step, the calculation of the utilizability. The daily total solar radiation Tl on the collector aperture is obtained by evaluating the formula
where Hh is the daily total hemispherical irradiation on the horizontal surrace, TJjTf^ is the monthly ratio of diffuse over hemispherical irradiation given by Eq. (3.6.1) and Fig. 3.6.1, and Rh and Rd are the functions listed in Tables 11.4.1-11.4.5. [The reader will recognize /?d and Rh as time integrals of rd and rh, Eqs. (3.6.4) and (3.6.5), multiplied by cos 0].9 The quantities a, b, and d in these tables are given by
9 The formulas for tracking collectors assume that the concentration is high enough to make the contribution of diffuse radiation negligible. More complete formulas that include diffuse radiation on tracking collectors can be found in Collares-Pereira and Rabl [1979].
310 TABLE 11.4.1
Active Solar Collectors and Their Applications Function Rh and Ra for Flat Plate Collector. pgrOLnd = Reflectance of Ground"
"From Collares-Pereira and Rabl [1979].
Rh and Rd depend on several hour angles, in particular, the sunset hour angle (or sunset hour ts):
given by
TABLE 11.4.2 Functions Rd and Rh for Concentrators with Fixed Aperture; e.g., Compound Parabolic Concentrator (CPC). C - Concentration"
"From Collares-Pereira and Rabl [1979],
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311
TABLE 11.4.3 Function Rh and Rd for a Collector That Tracks About East-West Axis"
Simpson's rule with one step will be adequate in most cases (error < 3%; for greater accuracy two steps can be used):
"From Collares-Pereira and Rabl [1979].
In the equations for flat plate and CPC there is also the angle u's given by
One further variable remains to be explained, the collector cutoff time tc, or equivalently, the cutoff angle
If the collector is placed due south (i.e., with zero azimuth) and if its time constant is short, it will operate symmetrically around solar noon, being turned on at 4 h before noon and turned off at tc h after noon. Obviously the largest possible value of tc is the sunset time 4. In some cases tc is further
TABLE 11.4.4 Functions Rh and Rd for Collector That Tracks About North-South Axis"
The integrals can be evaluated by Simpson's rule. "From Collares-Pereira and Rabl [1979],
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Active Solar Collectors and Their Applications
TABLE 11.4.5 Functions Rh and Rd for Collector with Two-Axis Tracking"
a
From Collares-Pereira and Rabl [1979].
limited by optical constraints, for example, shading. For CPCs the cutoff hour angle wc depends on the acceptance angle and is given by Eqs. (2.5.9) or (2.5.11). For collectors with heat loss, one proceeds to the second step of calculating the utilizability 0. The function <$> is denned in such a way that the energy Q delivered by a collector array is
K is the all-day average of the incidence angle modifier K(d) and can be approximated by
The utilizability is a function of the monthly average clearness index ~KT, the ratio
and the critical intensity ratio
Xc is the ratio of average daily heat loss and average daily absorbed solar radiation. For nontracking collectors is given by (for 0 < Xc < 1.2)
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System Models
For tracking collectors the R dependence can be neglected and the fit
can be used for all values of KT-£ 0.75 and for 0 < Xc < 1.2. For exceptionally clear climates, i.e., with KT > 0.75, the simple expression
should be used for all collector types. These curves are shown in Figs. 11.4.1-11.4.6. The fits were derived with emphasis on accuracy at reasonably large values of because collectors with low utilizability will not collect enough energy to be economical. The above expressions for are only reliable when (j> is larger than approximately 0.4 and when Xc < 1.2. Outside of this range these expressions for are not recommended. In fact, since the above fits may increase with Xc at large Xa they must not be used outside10 the range 0 < Xc< 1.2. The cutoff time tc for thermal collectors can be determined by the following simple iteration procedure, which is justified in Collares-Pereira and Rabl[1979a]. '°For flat plates larger critical intensity ratios can be explored with the correlations of [Klein, 1978]. The reader should be careful not to mix the notation used_by different authors. In particular 0 and H have different definitions; however, the product 0/fT of Klein is equal to our 4>H. Klein's definition of Xc is different.
Figure 11.4.1 Utilizability versus the critical ratio Xc for KI = 0.7 and nine values of R from 0-0.8 (From Collares-Pereira and Rabl [1979]).
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Active Solar Collectors and Their Applications
Figure 11.4.2 Utilizability <j> versus the critical ratio X,. for K-I- = 0.6 and nine values of R from 0-0.8 (From Collares-Pereira and Rabl [1979]).
(i) Start with tc — tc] = maximum permitted by optics, as discussed at the end of the previous section; for example, tcl = ts for flat plate or for tracking collectors if there is no shading. For the CPC, tcl is given by Eq. (2.5.9) or (2.5.11). (ii) Calculate corresponding output Q,. (iii) Decrease tc by A/ c to get new tc2 = tA — A/C(A^ = 0.5 h will give sufficient accuracy in most cases). (iv) Calculate output Q2 for tc2 and repeat procedure until maximal Q is found. The smaller the heat loss, the closer the optimal tc will be to tc[. This is
Figure 11.4.3 Utilizability versus the critical ratio Xc for K, = 0.5 and nine values of R from 0-0.8 (From Collares-Pereira and Rabl [1979]).
System Models
315
Figure 11.4.4 Utilizability 4> versus the critical ratio X, for K, — 0.4 and nine values of R from 0-0.8 (From Collares-Pereira and Rabl [1979]).
illustrated by the sample calculations in Tables 11.4.6 and 11.4.7 that were carried out with a rather small decrement &tc — 0.1 h. This small value was chosen only for the sake of illustration. For example, in Table 11.4.6 a value of A^. = 0.5 h would have yielded Q — 3.714 MJ/m2 on the second iteration, only 1% less than the value Q = 3.743 MJ/m2 obtained with A?c = 0.1 h on the 14th iteration. The maximum is broad and quite insensitive to uncertainties in tL., and a time step A/c = 0.5 h is adequate in practice. EXAMPLE 11.4.1
To provide an example, we calculate the energy delivery of several collector types for an average February day in New York, NY. The latitude is X = 40.5° and the sunset time ts = 5.24 h on February 15. The relevant values
Figure 11.4.5 Utilizability versus the critical ratio Xc for K, = 0.3 and nine values of R from 0-0.8 (From Collares-Pereira and Rabl [1979]).
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Active Solar Collectors and Their Applications
Figure 11.4.6 Utilizability versus the critical ratio Xc for collectors with high values of concentration and five KT from 0.3-0.7 (From CollaresPereira and Rabl [ 1979]).
of insolation Hh, clearness index Kr and daytime ambient temperature Tarab
are
TABLE 11.4.6 50°Ca
Collector Parameters and Energy Collected 15 February in New York at
1o#
U[W/m 2 °C] C Fm tc (h from noon) «*
Rd
R = Rd/Rh H [MJ/m 2 ][Eq. (11.4.1)] ^.[Eq. (11.4.12)] 0(#7-, «. ^) (GA4) = »o<*# [MJ/m2] rabs = 50°c
Flat plate
CPC evacuated
Parabolic trough EW tracking axis
0.75 4 1 0.9
0.6 0.8 1.5 0.99
0.65 0.7 20 0.95
3.934 1.626 0.783 0.481
4.651 1.723 1.089 0.632
5.234 1.874 1.932 1.031
5.234 2.377 2.549 1.072
5.234 2.441 2.617 1.072
10.616 0.697 0.470
10.278 0.213 0.807
8.385 0.237 0.856
10.274 0.194 0.881
10.550 0.054 0.966
3.743
4.976
4.667
5.886
6.624
3.369
4.926
4.434
5.592
5.962
(Ql A) = F,^007? [MJ/m2] ym = 50°c "From Collarcs-Perdra and Rabl [1979].
Parabolic trough polar tracking axis
2-axis tracker
0.65 0.7 20 0.95
0.65 0.2 500 0.9
317
System Models TABLE 11.4.7 Some Results of the Iteration Procedure to Determine the Cutoff Time Corresponding to the Maximum Energy (max (?) Collected" Iteration no. 1st 10th 14th (max. Q) 15th
Cutoff time
tc
R»
R«
H
Xc
Q
5.234 4.334 3.934 3.834
1.801 1.700 1.626 1.604
0.943 0.839 0.783 0.768
11.482 11.027 10.616 10.491
0.858 0.739 0.697 0.688
3.405 3.714 |3.743| 3.738
"Flat plate, tilt = latitude, at rabs = 50°C in New York in February as in Table 11.4.6. Iterations start with tc = ts = 5.234 sunset time (h from noon); decrement Arc = 0.1 h; for the sake of illustration Arc has been chosen much smaller than necessary (from Collares-Pereira and Rabl [1979]).
The correlation for the diffuse/hemispherical ratio, Eq. (3.6.1), yields a diffuse component of Hd = 3.78 MJ/m2 per day for these conditions. For the ground reflectance we assume pground = 0-2. We consider the following five collectors. (i) Flat plate collector with optical efficiency r/0Z = 0.75, U value U = 4.0 W/m2 °C, and heat extraction efficiency factor Fm = 0.90, typical of double glazing and selective coating. (ii) Fixed CPC collector with evacuated receiver, having TJOA" = 0.60, U = 0.8 W/m 2 °C, Fm = 0.99, concentration C - 1.5, and acceptance angle 26a = 68°. (iii) One-axis tracking concentrator with horizontal east-west axis and collector parameters ^K = 0.65, U = 0.7 W/m2 °C, and Fm = 0.95. (iv) Same collector as (iii) but with polar mount, i.e., tracking axis in north-south direction with tilt equal latitude. (v) Two-axis tracking Fresnel lens collector with ?j0 = 0.65, U = 0.2 W/ m2 °C, and Fm = 0.9. The operating temperature is specified as mean fluid temperature Tm = 50°C; this temperature is in the range appropriate for some space heating applications. The entries in Table 11.4.6 are obtained by iteration over tc with A^ = 0.1 h; details of the intermediate iterations are provided in Table 11.4.7. Rows 5 to 8 list the cutoff time tc, the functions Rh and Rd of Tables 11.4.1 to 11.4.5, and the ratio R = RJRh- The insolation H incident on the aperture during operating hours is obtained from Eq. (11.4.1) and listed in row 9. Row 10 lists the critical intensity ratio Xc of Eq. (11.4.12), and the corresponding value of the utilizability <j> of Eqs. (11.4.13) through (11.4.15) is entered in row 11. The final result for the delivered energy at !Tabs = 50°C is given in row 12 and at Tm - 50°C in row 13. At this operating temperature the heat loss from the flat plate is already so high that all the concentrating collectors in Table 11.4.6 outperform the flat plate. 11.5
THE , /-CHART
The correlations of Sections 11.3 and 11.4 can be used directly whenever the average operating temperature of the collector is known. For many sys-
318
Active Solar Collectors and Their Applications
terns with thermal storage the analysis is more complex because the collector temperature is not known a priori; rather, it fluctuates with the storage temperature which in turn depends on collector output during preceeding periods. Klein and Beckman [1979] have developed a shorthand procedure, the 0, /chart, for the analysis of the system shown in Fig. 1 1.1. la. This is essentially the schematic for an active space heating system; it could also be used for process heat (however, for the latter application, different designs are likely to be more efficient, as discussed in Section 12.3). Solar energy is supplied whenever the temperature of the storage tank is above Tmin, the minimum temperature useable by the load. The backup or auxiliary is in parallel with the solar system and supplies the entire load whenever storage falls below rmin. If the temperature of the collector were always equal to Tm-m (e.g., if storage were infinite), then the monthly delivered energy would simply be calculated according to Section 11.4, using the critical intensity ratio, Eq. (11.4.12),
for the collector inlet temperature rmin. The utilizability corresponding to XCi min serves as intermediate step in the $, /chart procedure. One calculates the variables
where L is the average daily load [in J]. If the system contains a heat exchanger between collector and storage, both X' and Y are multiplied by an additional factor Fx. [The numbers 100°C and 24 X 3600 sec in Eq. (11.5.3) have no fundamental significance; they were included by Klein and Beckman to make X' dimensionless.] With these values of X' and Fone can now determine how large a fraction/of the load the solar system will supply on the average. From the graphs in Fig. 11.5.1,/can be found using the X' and Y values as abscissa and ordiriate, respectively. The four storage capacities included in Fig. 11.5.1 cover the range of practical interest. The information in these figures is also contained in the equation
Rs is the ratio of the standard storage capacity (350 kJ/°C per m2 of collector) over the actual capacity. Equation (11.5.4) is implicit because/appears on both sides, but it can readily be solved by iteration. The difference between
System Models
319
y and / is the difference between finite storage and infinite storage. With finite storage Tm rises above 7^,, and hence /is lower than Y. With the , /chart method one calculation should be done for the central day of each month. The annual total delivered solar energy is then the sum of the monthly contributions
where Nt is the number of days and/L, the average daily solar contribution in each month.
EXAMPLE 11.5.1
A flat plate collector with ij0K =0.75 and ( 7 = 4 W/m 2 K is to supply pro cess heat at a minimum temperature Tm,n = 50°C. The location is New York, the collector tilt is equal to latitude, and the system schematic is as in Fig. 11.5.1. The average daily load is L = 430 MJ and the collector area is A = 100 m2. The storage capacity is 35 MJ/°K. Flow rate and heat exchanger are such that FmFx = 0.80. Calculate the solar energy delivered to the process, if storage is well insulated. SOLUTION
This is the flat plate used in Table 11.4.6, and the results can be used as basis for this calculation. The critical intensity ratio is the same, whether it is the mean fluid temperature Tm or the inlet temperature rHX, m on the storage side of the collector heat exchanger that is specified. Hence the calculation is the same as in Table 11.4.6, up to and including H = 10.616 MJ/m2 and
From Fig. 1 l.S.lb one finds/ = 0.60. This is a 14% reduction in delivered energy due to the rise of storage temperature above rmin. The solar energy delivered to the load during an average February day is NfL = 28.25 days X 0.60 X 430 MJ/day = 7.3 GJ. The , /chart assumes that the storage tank is well insulated and that there is no heat exchanger between storage and load. As for heat losses from the storage tank, they should be a small correction and thus only a rough estimate is needed. Usually the approximation
will be adequate. For a more accurate determination one can estimate the average collector inlet temperature rin by backtracking the utilizability cal-
320
Active Solar Collectors and Their Applications
Figure 11.5.1 , /charts for various storage capacities (a) Storage capacity 175 KJ/°C nr; (b) Storage capacity 350 kJ/°C nr; (c) Storage capacity 700 kJ/°C nr; (d) Storage capacity 1400 kJ/°C m2 (From Klein and Beckman [1979]; reprinted by permission of John Wiley & Sons).
culation. For this purpose one replaces Fin Eq. (11.5.2) by the value of/ that has been obtained for the month, and then one calculates the corresponding value of the utilizability:
Knowing , one can use Eqs. (11.4.12) through (11.4.15) to solve for Xc,m and hence Tin. Klein and Beckman [1979] recommend that the arithmetic average of Tmm and Tm be used instead of Tmir, in Eq. (11.5.6)." "A more accurate correlation for tank Josses has recently been presented by Braun et al. [1982].
System Models
Figure 11.5.1
321
(continued)
If there is a heat exchanger between storage and load, Duffie and Beckman [1980] assume that Tmm must be increased by
where L is the average load rate [in W], eL is heat exchanger effectiveness, and (mc)mm is the smaller of the two fluid capacity rates in the heat exchanger.
322
Active Solar Collectors and Their Applications
EXAMPLE 11.5.2 The storage tank for the previous example has a loss coefficient-area product (UA)Mr = 6 W/K and is placed in an environment with Tmv — 12°C. What is the loss from storage in February? SOLUTION If one assumes Eq. (11.5.6) one finds
approximately an 8% loss compared to 7.3 GJ. Let us see how much this estimate changes if we take into account the temperature rise above Tmm. Comparing Eqs. (11.5.2) and (11.5.7) we find
and hence 4>(X^n) = 0.403. From Fig. 11.4.4 we read Xc-m = 0.84. Since X.in/X.min = (tin - l\mb)/(Tmm - T.dmb) one obtains Tm = 60°C. With the new storage temperature estimate of (50°C — 60°C)/2 = 55°C, the storage loss estimate increases to 0.63 GJ. This increase amounts to about 1% of the delivered energy and is not significant in view of the overall errors. Hence the simple estimate, Eq. (11.5.6), gives an acceptable answer for the storage tank loss in this case.
REFERENCES Alcone, J. M. and Herman, R. W. 1981. "Simplified Methodology for Choosing Controller Set Points." ASME 3rd Conference on System Simulation, Economic Analysis, Solar Heating and Cooling Operational Results. Reno, NV. Braun, J. E., Klein, S. A., and Pearson, K. A. 1982. "An Improved Design Method for Solar Water Heating Systems." University of Wisconsin Report. Beckman, W. A., Klein, S. A., and Duffie, J. A. 1977. Solar Heating Design by thefchart Method. New York: John Wiley & Sons. Boes, E. C. et al. 1977. "Availability of Direct, Total and Diffuse Solar Radiation to Fixed and Tracking Collectors in the USA." Report SAND-77-0885. Sandia Laboratories. Bruno, R. 1979. "Which Models for What?" Philips GmbH Forschungslaboratorium Aachen, Report No. 7/79. In Proc. Joint German-Australian Workshop (CCMS) on Solar Energy System Design. Collares-Pereira, M. and Rabl, A. 1979a. "Derivation of Method for Predicting Long Term Average Energy Delivery of Solar Collectors." Solar Energy 23:223. Collares-Pereira, M. and Rabl, A. 1979b. "Simple Procedure for Predicting Longterm Average Performance of Nonconcentrating and of Concentrating Solar Collectors." Solar Energy 23:235.
System Models
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Dickinson, W. C. 1978. "Annual Available Radiation for Fixed and Tracking Collectors." Solar Energy 21:249. Dudley, V. E. and Workhoven, R. M. 1979. Summary Report: Concentrating Solar Collector Test Results Collector Module Test Facility. Report SAND-78-0977. Albuquerque, NM: Sandia Laboratories. Duffie, J. A. and Beckman, W. A. 1980. Solar Engineering of Thermal Processes. New York: John Wiley & Sons. Fewell, M. E. and Grandjean, N. R. 1979. "User's Manual for Computer Code SOLTES-1 (Simulator of Large Thermal Energy Systems)." Report SAND 781315. Albuquerque, NM: Sandia Laboratories. Gaul, H. W. and Rabl, A. 1980. "Incidence Angle Modifier and Average Optical Efficiency of Parabolic Trough Collectors." Trans. ASME J. Solar Energy Eng., 102:16. Gordon, J., and Zarmi, Y. 1982. "A Simple Method for Calculating Annual Insolation on Solar Collectors." Solar Energy 28:483. Gordon, J. M. and Rabl, A. 1982. "Design, Analysis and Optimization of Solar Industrial Process Heat Plants Without Storage." Solar Energy 28:519. Gordon, J. M. and Zarmi, Y. 1983a. "The Utilizability Function: Theoretical Development of a New Approach." Solar Energy 31:529. Gordon, J. M. and Zarmi, Y. 1983b. "The Utilizability Function: Validation of Theory Against Data-Based Correlations." Solar Energy 31:537. Hall, I. J. et al. 1978. "Generation of Typical Meteorological Years for 26 SOLMET Stations." Report SAND 78-1601. Albuquerque, NM: Sandia Laboratories. Hottel, H. C. and Whillier, A. 1955. "Evaluation of Rat Plate Solar Collector Performance." Transactions of the Conference on the Use of Solar Energy: The Scientific Basis, Vol. II, Part 1, Section A, pp. 74-104. Klein, S. A. et al. 1978. "Calculation of Flat-Plate Collector Utilizability." Solar Energy 21:393. Klein, S. A. et al. 1979. "TRNSYS-A Transient Simulation Program; Users Manual." Engineering Experiment Station Report 38. Madison, WI: Solar Energy Laboratory, University of Wisconsin. Klein, S. A. and Beckman, W. A. 1979. "A General Design Method for Closed-Loop Solar Energy Systems." Solar Energy 22:269. Kusuda, T. 1981. "A Comparison of Energy Calculation Procedures." ASHRAE J. August:21. Kutscher, C. F., Davenport, R. L., Dougherty, D. A., Gee, R. C., Masterson, P. M., and May, E. K. 1982. "Design Approaches for Solar Industrial Process Heat Systems." Report SERI/TR-253-1356. Golden, CO: Solar Energy Research Institute. Liu, B. Y. H. and Jordan, R. C. 1963. "A Rational Procedure for Predicting the Long Term Average Performance of Flat-Plate Solar-Energy Collectors." Solar Energy 7:53. Lunde, P. J. 1980. Solar Thermal Engineering. New York: John Wiley & Sons. Mitchell, J. D., Theilacker, J. C., and Klein, S. A. 1981. "Calculation of Monthly Average Collector Operating Time and Parasitic Energy Requirements." Solar Energy 26:555. Rabl, A. 1981. "Yearly Average Performance of the Principal Solar Collector Types." Solar Energy 27:215. Sicgel, M. D., Klein, S. A., and Beckman, W. A. 1981. "A Simplified Method for
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Estimating the Monthly Performance of Photovoltaic Systems." Solar Energy 26:413. SOLMET. 1978. "SOLMET, Volume 1-User's Manual TD-9724, Hourly Solar Radiation-Surface Meteorological Observations." Asheville, NC: National Climatic Center. Zarmi, Y. 1982. "Transition Time Effects in Collector Systems Coupled to a WellMixed Storage." Solar Energy 29:1.
12. APPLICATIONS
In this chapter we discuss the principal uses of active solar collectors: space heating and/or cooling, water heating, industrial process heat, and power generation. For some of the most common applications explicit design procedures are presented. A design procedure consists of the following steps: (i) selection of system configuration (flow diagram) (ii) selection of manufacturer, model, and size for each of the system components (collector, storage, etc.) (iii) calculation of system performance (iv) economic analysis (v) variation of size and model selection for each system component to determine economic optimum The design procedures in this chapter deal with the first three stages, in particular the calculation of system performance. The economic analysis and optimization, described in Chapters 14 and 15, are the same for all applications. 12.1 ACTIVE SPACE HEATING AND COOLING There are basically two approaches to space heating with solar energy. Active solar heating employs mechanical equipment (e.g., collectors, pumps, ducts, pipes, storage tanks) that is separate from the architecture of the building. The passive approach, on the other hand, relies on parts of the building itself (e.g., windows, storage walls, atrium) to collect and store solar heat. Sometimes one may also combine active and passive elements in a hybrid design. This book deals only with active systems. For passive solar heating the reader is referred elsewhere; for instance, Kreider and Kreith [1982], Balcomb [1980], and Anderson and Michal [1980]. In most regions active solar space heating is a challenging task because the load tends to be highest when there is least insolation. To compensate as much as possible for this mismatch, one tilts the collectors towards the low winter sun; in midlatitudes a good rule of thumb gives the collector tilt as latitude plus 15°. During summer the collectors are idle, and one may have to take precautions to keep the system from overheating. As an alternative one could use seasonal storage from summer to winter, for example, large water tanks [Sillman, 1981] or deep solar ponds [Rabl and Nielsen,
325
326
Active Solar Collectors and Their Applications
1975]. For a given space heating demand, the seasonal storage approach requires less collectors but much more storage than the conventional active system with short term storage. For conventional systems one recommends on the order of 300 kJ/°C m2 of thermal storage capacity per collector area; this is enough for at most a few cloudy days. The heat available during summer could be utilized for solar cooling by means of an absorption chiller or a desiccant chiller. 12.1.1 System configurations The two main classes of active space heating systems are distinguished by the heat transfer fluid in the collector: liquid systems and air systems. Schematics for each are shown in Figs. 12.1.1 and 12.1.2. In general, heat transfer to air is poor. Nonetheless solar air systems may be practical, if the heat distribution system of the building uses air anyway. In that case at least one heat exchange between collector and air is necessary, and that heat exchange might as well take place in the collector rather than in a separate heat exchanger. For this reason the overall system performance of an air based system is comparable to that of a liquid system despite the seemingly lower collector efficiency (as reflected in lower values for the heat transfer factors Fm and Fm of the collector efficiency equation). Typical design parameters are listed in Table 12.1.1 for liquid and in Table 12.1.2 for air systems.
Figure 12.1.1 Typical system schematic for liquid space heating system (From Kreider and Kreith [1982]).
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327
Figure 12.1.2 Typical system schematic for air space heating system (From Kreider and Kreith [1982]). 12.1.2
Performance prediction for space heating
Many computer programs and shorthand procedures have been developed for predicting the performance of solar heating systems; some of them are described by Kreider and Kreith [1982] and by Bendt and Soto [1980]. In this chapter we present the /chart method of Beckman, Klein, and Duffie [1977]. This method has been derived as curve fits to the results of a large number of computer simulations. It requires one calculation per month. TABLE 1 2. 1 . 1 Typical Design Parameter Ranges for Liquid Solar Heating Systems" Collector flow rate Collector slope Collector azimuth Collector heat exchanger Storage capacity Load heat exchanger Water preheat tank capacity
0.0 10-0.020 kg/seem2 (\ + 15°) ± 15° 0° ± 15° F, > 0.9 50-100 liters/m2 1 < f(mc)min/L < 5 1.5 X capacity of conventional heater
"From Duffie and Beckman [1980]. Reprinted by permission of John Wiley & Sons.
328
Active Solar Collectors and Their Applications TABLE 1 2. 1 .2 Typical Design Parameter Ranges for Solar Air Heating Systems3 Collector air flow rate Collector slope Collector surface azimuth angle Storage capacity Pebble size (graded to uniform size) Bed length, flow direction Pressure drops Pebble bed Collectors Ductwork Maximum entry velocity of air into pebble bed (at 55 Pa pressure drop in bed) Water preheat tank capacity a
5-20 liters/m2 sec (A + 15°) ± 15° 0° ± 15° 0.1 5-0.35 in3 pebbles/m2 0.01-0.03 m 1. 25-2.5 m 55 Pa minimum 50 to 200 Pa 10 Pa 4 m/sec 1.5 X conventional water heater
From Duffic and Beckman [1980]. Reprinted by permission of John Wiley & Sons.
The input variables of the system enter the calculation in two dimensionless groups:
and
where A Fm Fx U ramb Tday
L j]0 K H
= = = = = = = -
collector area [m2] heat transfer factor heat exchanger factor collector heat loss coefficient (U value) [W/m2 K] monthly average ambient temperature [C°] 24 X 3600 sec total daily heat load [J] averaged over the month optical efficiency all-day average of incidence angle modifier [Eq. (11.4.10)] daily solar irradiation incident on collector [J/m2] [Eq. (11.4.1) with cut off time tc = ts]
The variable Xf represents the collector heat loss, the variable Yf the absorbed solar radiation; both are normalized by the heating load L. One wants to find / the fraction of the load L that is supplied by solar energy. The quantity /is a function of Xf and Yf and is given both in equation and graph form. For liquid systems/is given by Fig. 12.1.3 or by the equation
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Applications
Figure 12.1.3 The/-chart for systems with liquid heat transfer and storage media (From Beckman, Klein, and Duffie [1977]; reprinted by permission of John Wiley & Sons).
For air systems one uses Fig. 12.1.4 or the equation / = 1.040}}- 0.0651}- 0.1591^ + 0.00187^ - 0.0095Y}.
(12.1.4)
These equations should not be used outside the range shown by the curves in Figs. 12.1.3 and 12.1.4. This procedure is repeated for each month of the heating season. The annual delivered solar heat Q is the sum of the monthly contributions:
where Nt is the number of days and /L, is the average daily solar contribution in each month. The/chart correlations are based on standard storage capacities 75 kg of water per m2 of collector for liquid systems and 350 kJ/°C per m2 of collector for air systems. For other storage capacities one simply corrects the variable
Figure 12.1.4 The/chart for air systems (From Beckman, Klein, and Duffie [1977]; reprinted by permission of John Wiley & Sons).
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Active Solar Collectors and Their Applications
Xfby the correction factor
with the exponent
This correction is valid for the range 0.5 <
< 4.0.
In general, /chart results agree within a few percent with the results of detailed computer simulations, in particular, the TRNSYS model from which the /chart was developed. 12.1.3 Measured system performance Several solar heated houses have been instrumented and monitored. Examples are the MIT solar houses [Engebretson, 1964], the Lof residence [Lof et al., 1964], and the Colorado State University solar houses [Karaki et al., 1977; Karaki et al., 1978]. Performance data for one of these, the Colorado State University Solar House II, are summarized in Table 12.1.3. Of particular interest are the system efficiency and the solar fraction. The monthly average system efficiency, based on total insolation incident on the collector, is on the order of 25%-30% during the heating season. The solar fraction varies with time of year from about 0.6 to about 0.9. These numbers are quite typical of active solar heating systems. The performance data for this house have been used to validate the /chart procedure. The measured annual solar heating fraction was 0.72 while/chart predicted 0.76 [Duffi and Beckman, 1980]. The agreement is really quite good, considering how many reasons there could be for a discrepancy. If the system, as actually built and operated, had some defect (e.g., leaks or control failures) the performance would be lower than predicted by /-chart, even if the latter were perfectly accurate. For a recent survey of systems with measured performance and a comparison to/chart predictions the reader is referred to Duffle and Mitchell [1983]. More recently measured performance data have become available for a large number of solar heating and cooling systems, as published in an impressive series of reports by Vitro Labs [1981], 12.1.4 Solar cooling Solar energy could provide cooling by several techniques, the most important being the absorption chiller, the desiccant chiller, and the compression chiller. The first two use solar heat directly while the latter operates on mechanical power from a photovoltaic or a solar thermal generator. Desic-
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TABLE 12.1.3 Thermal Performance Data for a Solar Space Heating System (Air SystemColorado State University Solar House II)a 1977
1976
Collector (MJ/m 2 day): Total solar insolation Solar insolation while collecting Heat collected Efficiency (percent) Based on total solar insolation Based on solar insolation while collecting Space and DW Heating (MJ/day): Total energy required Space
DHW Total Solar contribution Space
DHW Total Solar fraction Space
DHW Total
Nov. 24 days
Dec. 10 days
Jan. 26 days
Feb. 20 days
Mar. 27 days
Apr. 18 days
13.5
15.8
15.1
15.6
17.7
14.2
11.2
13.8
12.9
3.1
4.9
4.6
13.6
15.6
12.2
5.0
3.7
23.0
31.0
30.0
30.0
28.0
26.0
28.0
36.0
36.0
35.0
32.0
30.0
311.6 40.6 352.2
323.2 52.3 375.5
432.9 73.0 505.9
319.9 82.9 402.8
319.0 75.4 394.4
192.3 62.8 255.1
198.7 21.3 220.0
257.5 30.8 288.5
231.4 63.5 294.9
239.6 75.4 315.0
245.8 67.4 313.2
166.1 51.8 217.9
0.64 0.52 0.62
0.80 0.59 0.77
0.53 0.87 0.58
4.7
0.75 0.91 0.78
0.77 0.89 0.79
0.86 0.82 0.85
"From Karakietal. [1977],
cant cooling is still in the development stage, and it may hold promise. Absorption cooling is a well developed technology. Of the two cycles in common use, the lithium bromide cycle is usually operated at 100-120°C and is thus better matched to solar collectors, whereas the ammonia water cycle requires temperatures around 150-180°C. Lithium bromide chillers can be modified to run on heat at temperatures as low as 85°C, but then their capacity is reduced. Energy storage with solar absorption chillers is fairly costly because only a small temperature interval is available, whether the energy is stored as chilled water or as hot water. Furthermore, the transient operation typical of solar energy systems imposes penalties. To evaluate the prospects of solar cooling with compression chillers, it is helpful to think of this approach in two separate steps: first power is generated, then that power is used for cooling. These two processes are really independent of each other. Therefore one might as well compare the following two systems for generating power: (i) a compression chiller driven by mechanical power from a flat plate solar thermal system on a building, and (ii) the same chiller driven by electricity from a central station high temperature solar thermal power plant. Not only does the local system lack economies of scale, but with a flat plate system the conversion efficiency to elec-
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Active Solar Collectors and Their Applications
tricity would be far too low, as shown by the example in Table 12.5.2. The comparison is so unfavorable for local solar thermal power generation that it is not substantially altered if one were to use evacuated tubes or other medium-temperature collectors that could reasonably be deployed on an individual house. Thus one should not expect to improve the cost effectiveness of a solar heating system by adding solar cooling with compression chillers. When considering compression chillers, the real question is: what is the most appropriate technology for generating power? Perhaps the best cooling technologies with renewable energy are not based on active solar approaches. For example, in dry climates evaporative cooling seems to be the most economical [Supple, 1982]. In dry climates radiative cooling at night can also be very effective [Hay and Yellot, 1970]. In humid climates with cold winters the ice pond may be cost effective for large applications such as commercial buildings; using a snow machine, one makes ice/snow in winter and stores it for use in summer [Kirkpatrick et al., 1983]. 12.2 WATER HEATING The heating of domestic or service water, water heating in short, is one of the oldest applications of solar energy. In the time from 1920-1950 thousands of solar water heaters were in use in California and Florida [Butti and Perlin, 1980]. They fell out of favor when conventional energy became cheap, but since the energy shortage of the seventies they have reappeared in ever increasing numbers. In some countries with plentiful sunshine and high energy prices, most notably in Israel, solar water heaters are very cost effective and popular. There are two good reasons why solar energy is well suited for water heating. The temperature range is well matched to inexpensive flat plate collectors, and the demand for hot water is quite uniform throughout the year. 12.2.1 System types There are several possible ways of connecting solar collectors to a water supply system. The most important solar water heater configurations are shown in Fig. 12.2.1. The first two systems Figs. 12.2.la and 12.2.Ib are called direct because the water makes direct contact with the collector, by contrast to the indirect systems of Figs. 12.2.1c and 12.2. Id, where a heat exchanger separates the collector loop from the water supply. Typically one uses a water-antifreeze mixture in the collector loop. Because of the toxicity of most commercial antifreezes (glycol), U.S. building codes require a double wall heat exchanger in that case. Obviously a heat exchanger increases the cost and it imposes a performance penalty on the order of a few percent. However, in locations with frost the indirect systems may be the most desirable choice, and they are the most popular ones in the U.S. Also, direct systems may be problematic in areas where the pressure of the water mains
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333
Figure 12.2.1 Schematics of common configurations of water heaters, (a) A natural circulation system, (b) One tank forced circulation system, (c) System with antifreeze loop and internal heat exchanger, (d) System with antifreeze loop and external heat exchanger. Auxiliary is shown added in the tank, in a line heater, or in a second tank; any of these auxiliary methods can be used with any of the collector-tank arrangements. (From Duffie and Beckman [1980]; reprinted by permission of John Wiley & Sons).
exceeds the rated pressure limit of the collector or where water hardness is excessive. Frost protection for direct systems requires special care. In regions where frost is light and rare, one can afford to heat the collector artificially when necessary, by circulating a small amount of warm water through it. One particularly elegant design uses a valve that opens whenever the ambient approaches freezing temperatures. When the valve opens, cold water is allowed to drain slowly out of the collector into the open, and warmer water from tank or mains enters the collector as replenishment until this warmer water reaches the valve and closes it again. The cost of the discarded water is negligible. The valve is a simple passive device based on the differential thermal expansion of dissimilar materials. In regions with more severe frost one needs to drain the collector to prevent freeze-ups. In the U.S. the draindown approach has been taken most frequently when a direct system has been used at all. In principle the design of a draindown system is straight-
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Active Solar Collectors and Their Applications
forward, requiring only valves that are activated when the absorber temperature sinks below a specified value. However, in practice there are problems with reliability.' A single failure may destroy the entire collector and can be more costly than the cost and performance penalties of indirect systems. The simplest system is the thermosyphon in Fig. 12.2. la. Neither pump nor controls are needed because the fluid circulates by virtue of its own thermal expansion whenever the temperature in the collector is above the temperature at the top of the storage tank. The flow rate depends on the flow resistance in the collector loop. As shown by a theorem of Gordon and Zarmi [1981] and by the computer simulations of Mertol et al. [1981], the flow rate has little effect on daily average performance, provided the flow rate is high enough to move the entire storage volume through the collector at least once during a sunny day. This conclusion about flow rate independence holds for pumped systems just as much as for thermosyphons. As a completely passive system the thermosyphon is the most efficient and the most reliable. However, it can only be used if certain conditions are satisfied. Apart from the above-mentioned problem of frost protection, the storage tank must not be located below the collector. Since the collector is usually mounted on the roof of a building, this requirement may be unacceptable on architectural grounds. Also, a storage tank on the roof may require special protection against freezing. It is possible to combine the simplicity of a passive thermosyphon with the fail-safe free/e protection of an indirect system by adding a heat exchanger. However, the requirement for placing the storage tank above the collector still remains. If the tank must be located below the collector, then the fluid must be pumped actively through the collector. This necessitates not only controls, as indicated in Fig. 12.2.1b, but also a check valve to prevent thermosyphon flow at night from storage back to the collector. The indirect system of Fig. 12.2. Ic differs from Fig. 12.2.1b only by the addition of a heat exchanger. Finally, Fig. 12.2. Id shows an indirect system with two tanks. In a two-tank system the solar components act as preheater to the conventional system. Much of the time the temperature in the preheat tank is below the water use temperature, and hence the collector can operate at higher efficiency than in a single tank system. In practice this advantage is more or less canceled by losses from the increased tank surface. In onetank systems one can capture at least some of the benefit of preheating by placing the auxiliary heater at the top of the tank while solar heat is added only at the bottom of the tank, as indicated by the placement of the lines in Fig. 12.2.1. For improved performance one can place the auxiliary tank directly on top of the solar preheater tank, with a thermal diode (e.g., a heatpipe) between the tanks. Such a system has been developed by Grunes et al. [1982]. 'Also, some collectors have serpentine flow paths and do not drain well.
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Applications
12.2.2 Performance comparison
Several of the standard solar water heating systems have been evaluated experimentally in a study by the National Bureau of Standards [Fanney and Liu, 1979; Farrington et al., 1980]. Six systems were operated side by side, exposed to the same climatic conditions and supplying approximately the same thermal load. The systems were selected as typical of those being installed commercially. The most important performance results are summarized in Table 12.2.1, in terms of the thermal efficiency of the collector, system efficiency, and solar fraction. Before interpreting the results, one should keep in mind that the single-tank direct system and the air system had different collector areas from the other four systems. The system efficiency can be significantly lower due to parasitic power requirements and due to losses from the tank. Since parasitic power is supplied as electricity from a conventional central station power plant, the system efficiency and net solar fraction have also been calculated considering the fossil fuel energy consumed at the power plant; the latter numbers are given in parentheses in Table 12.2.1. The following conclusions are suggested. Air systems have such a high airto-water heat exchanger penalty and require so much parasitic power as to be impractical. For liquid systems the collector efficiency is in the range of about 23%-35% while the system efficiency is in the range of 18%-29%. Taking into account the fuel consumed by the central station power plant for the parasitic energy, the system efficiency is in the range of 8%-24%. Thermosyphon systems and single-tank direct systems have the highest efficiency. (The higher collector efficiency of the single-tank direct system in Table 12.2.1 is probably due to the smaller collector area, which results in lower solar fractions and lower tank temperatures.) The system efficiency for indirect systems and for double-tank systems is significantly lower. When one considers systems costs in addition, one finds that the cost of solar energy for indirect and for double-tank systems is likely to be 2 to 3 times higher than for the thermosyphon system. TABLE 1 2.2. 1 Performance Results for Six Water Heater Systems"
System
Collector area (m2)
Thermal efficiency (%)
System efficiency (%)
Solar fraction0
Net solar fraction
Thermosyphon Single, dir. Single, ind. Double, dir. Double, ind. Air system
5.0 3.3 5.0 5.0 5.0 7.3
26.4 35.3 24.7 23.3 22.5 11.5
25.7 (24.3)b 28.5 (14.9) 22.3(17.5) 18.1 (7.7) 20.0(15.4) 8.1 (1.30)
0.57 0.51 0.54 0.52 0.50 0.38
0.56 (0.52) 0.42 (0.22) 0.49 (0.38) 0.41 (0.17) 0.44 (0.33) 0.26 (0.03)
"Adapted from Farrington et al. [1980]. b Figures in parentheses represent values if parasitic energy consumption were considered as energy required at a fossil-fueled electric generating plant (33% electric plant efficiency assumed). c Collector areas must be considered when comparing solar fractions.
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Active Solar Collectors and Their Applications
12.2.3 Performance prediction for water heaters The performance of solar water heaters can be predicted by the /chart method described in Section 12.1.2 for space heating. Only one slight modification is required to adapt the /-chart for water heating. The variable Xfi which accounts for collector heat losses, needs to include a correction for the temperature rmains of the main water supply and for the minimum acceptable hot water temperature Tlmd since both of these have an effect on the collector operating temperature. Duffie and Beckman [1980] recommend the following expression for the variable Xf.
where all temperatures are in °C. The other symbols and the variable I/are the same as in Section 12.1.2. The solar fraction/is given by the same function of Xf and 7/as for liquid space heating systems, i.e., Eq. (12.1.3) and Fig. 12.1.3. The calculation is to be repeated for each month of the year. The/chart correlation for water heating is based on the two-tank configuration of Fig. 12.2. Id. It assumes a standard storage capacity of 75 kg/m2 of collector area. The effect of different storage sizes depends on the load pattern and is difficult to predict. The tanks are assumed to be well mixed and the energy in water above r,oad is considered not to be useful. The latter two assumptions yield conservative predictions. Heat losses from the auxiliary tank are to be included as part of the load (as they are normally included in the energy supplied in a conventional water heater). EXAMPLE 12.2.1
Calculate the useful solar heat supplied by a flat plate system in Denver, CO, in March. The collector is deployed at tilt equal latitude and receives an average irradiation of JJ = 21.6 MJ/m2 per day. The aperture area is A = 6 m2 and the storage capacity has the standard value of 75 kg/m2 of collector area. The collector-plus-heat exchanger combination has the parameter FxFmrj0K = 0.65 and FxFmU = 3.6 W/m2 K. The hot water demand is 400 kg/day at r,oad = 50°C. The temperature of the_water supply is Tmains = 10°C, and the average ambient air temperature is ramb = 3°C. To estimate heat losses from the auxiliary tank, assume a value (UA)stor = 1.2 W/K and a tank environment at 7"env = 20°C. SOLUTION
The daily hot water load is
Applications
337
To this, one has to add the heat loss from the auxiliary tank:
Thus the total daily load is
The variable Xffor the/chart is, from Eq. (12.2.1),
The variable 7, is, from Eq. (12.1.2),
From Eq.(12.1.3) or Fig. 12.1.3 we obtain the solar fraction
Hence this solar system saves
12.2.4 Swimming pool heaters Swimming pools are an obvious candidate for solar heating because of the low temperature required. In many cases the cheapest collector, a simple unglazed flat plate, will turn out to be the most cost-effective choice for an active pool heater. However, one should not overlook the following passive alternative. If an outdoor pool is used only during a few hours of the day, it may be best to cover it with a transparent insulating cover for the rest of the time. Such a cover can be made of a double plastic film with an enclosed air layer; it reduces heat losses while admitting solar radiation. A detailed evaluation is necessary in each case to determine which approach is best. To evaluate a solar pool heater one needs to calculate both the delivered solar heat and the heat loss from the pool. The temperature of a swimming
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Active Solar Collectors and Their Applications
pool during the swimming season will be within a rather narrow comfort range; furthermore the heat capacity is so large that the temperature change during the day has negligible effect on the efficiency of a solar collector. Therefore one can employ the monthly utilizability method of Section 11.4 for calculating the useful solar heat, as pointed out by Govaer [1984]. The calculation of the load is more complicated, and the procedure of Govaer and Zarmi [1981] can be used. 12.3 INDUSTRIAL PROCESS HEAT 12.3.1 General considerations Heat is used in a wide variety of industrial processes. As long as solar equipment is relatively expensive one needs to select the applications carefully to obtain a good match between a process and a solar system. The higher the temperature, the lower the efficiency of the collector. Hence one should take advantage, as much as possible, of opportunities to supply heat at low temperatures. This necessitates a close examination of the process characteristics. Consider, for example, a plant where a central steam distribution system is used to heat a water bath; in such a case a solar retrofit could provide hot water directly to the bath rather than high-temperature heat to the central boiler. In many process heat systems a process fluid is heated over a range of temperatures; a relatively small collector field used as preheater can operate more efficiently than a large field that supplies the entire load. In theory the possibility of small and highly efficient solar preheaters exists also in domestic hot water applications, but in practice such systems would be too small to be economical. Industrial loads, on the other hand, are so large that even a solar preheater is large enough to reap the benefit of economies of scale. In the near term it is thus advisable to limit solar applications in industry to those cases that are most favorable by virtue of temperature, load distribution, etc. In the future as research, development, and growing mass production reduce the cost of solar installations, an ever larger number of applications will become economically attractive for solar. Of course, in evaluating the use of solar energy for an industrial application, one should not overlook opportunities for energy conservation. Frequently, solar equipment and energy conservation measures compete directly with each other. A heat recovery unit, for instance, will save low-temperature heat, thereby eliminating some of the best solar potential. As with all capital intensive technologies the load distribution is crucial. Most industries operate at a more or less uniform rate. Capital intensive industries tend to run continuously, 24 h/day, 7 days/week, while labor intensive industries are likely to shut down for nights and weekends. Other load patterns (e.g., 16 h/day, 6 days/week) also occur. It is easy to achieve good utilization of solar equipment in an industry with constant load 7 days/week. One simply chooses the size so that the peak output of the solar
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system will just meet the load. The situation is more difficult if the industry shuts down for the weekend. In that case one either discards the solar energy collected during weekends or one adds storage. Because of differences in load distribution, the role of solar storage for industry differs from residential applications. Several distinct uses of storage are possible. Buffer storage for short periods (on the order of 1 h) may be necessary if the process is discontinuous (batch process) or if the backup cannot respond quickly to variations of solar input. Day/night storage permits use of solar heat at night. Weekend storage is an option for factories that close for the weekend. Finally there is the possibility of long term storage. Each of these requires different storage capacities. Which type of storage one wants, if any, depends on the characteristics of a particular application. In fact, in many situations, a design without storage may be preferable. After all, most industries have a load while the sun is shining. Thus, solar energy need not be stored to be useful (by contrast to solar domestic hot water, which must be stored because the load is out of phase with the sun). Storage increases the cost of solar energy because of added cost of the storage tank and because of losses from storage. Therefore the least expensive solar industrial process heat systems will contain minimal storage or none at all, and they will use fossil fuel backup for times when the amount of sunshine is insufficient. In view of this, one can distinguish the following three possibilities: 1. solar without storage costs more than fossil fuel 2. solar without storage costs less, and solar with storage costs more than fossil fuel 3. solar with storage costs less than fossil fuel In case (1) solar cannot compete. In case (2) one will build a solar system without storage that provides only the daylight portion of the load. Finally in case (3) the optimal system contains storage and supplies a large portion of the total load. The design of a solar industrial process heat system depends on the use of storage. Accordingly we discuss several different designs. The system of Section 12.3.2 contains no storage; it is the simplest possible solar energy design and the most efficient. The single-pass system of Section 12.3.3 contains storage and is nearly as efficient as the no-storage system. However, its application is somewhat limited because it is essentially an open-loop design. In practice it is very well suited for hot water loads. For high-temperature applications (e.g., with oil as heat transfer medium), it would require two separate storage tanks. For air systems it does not appear practical. Finally the multiple-pass system with storage, Section 12.3.4, can be used for all applications, but its efficiency is significantly below that of the first two designs. The formulas we present for the single-pass designs are for the most important case of a load that is constant in temperature and flow rate. If the heat rate of the process varies with time, it may be most economical to match the solar system to the constant base component of the load rather than to the total load. If solar equipment is sufficiently cheap, one may want to choose a larger system with added stor-
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Active Solar Collectors and Their Applications
age to bridge load variations. If the load varies with season, a monthly calculation based on the monthly utilizability model of Section 11.4 is needed. The basic design philosophy of the single-pass open-loop approach still applies. The formulas also hold for the case of plants with batch operation, provided a buffer storage is included to make the load essentially continuous. 12.3.2 Systems without storage for constant loads For the no-storage constant load system, the simplest possible flow diagram is shown in Fig. 12.3.1. The collector is in series with backup and load. The process requires fluid at a temperature r,oad and the return temperature from the process is the inlet temperature Tm of the collector field. The flow rate m through the collector field equals the process flow rate m,oad. Flow rate and collector inlet temperature are constant. [Gordon and Rabl, 1982]. The load rate L [in W] is
Even though Fig. 12.3.1 shows a closed loop, it covers the open-loop case as well, e.g., a system where water from the water mains or ambient air is pumped through the collector and discarded after the process. For the purpose of the analysis, an open-loop is equivalent to a closed loop where the
Figure 12.3.1 Flow diagram for industrial process heat system without storage, (a) Direct, (b) With heat exchanger.
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return temperature from the process equals the temperature of the water mains or of ambient air. In some cases the process fluid may be incompatible with the collector, and a heat exchanger must be interposed between collector and process. The effect of the heat exchanger can be accounted for very easily by multiplying the instantaneous efficiency equation by the heat exchanger factor Fx. The starting point of the analysis is the instantaneous efficiency equation of the collector field
with
The collectors are turned on if and only if the solar irradiance on the collector aperture exceeds the threshold X:
Under these conditions the yearly heat Q delivered by the collector field is given by
where q(X) is the correlation
displayed in Figs. 11.3.1-11.3.7. This equation gives the yearly useful energy as long as no collected heat is discarded. However, as shown in Chapter 1 5, the economic optimum i likely to correspond to areas large enough to incur some energy dumping. Hence we need to generalize the expression for Q to include energy dumping. As the first step let us calculate the area at which dumping begins. Let us call this value Adump. It is determined by the simple requirement that the power delivered under peak insolation just match the load rate L. The peak insolation for the collectors under consideration is, for midlatitudes near sea level, 0.85-1.10 k W/m for flat plates 0.801.00 k W/m for concentrators,
the range of numbers corresponding to the range from cloudy to clear climates. (More precisely 7max is equal to that value of X at which the correla-
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Active Solar Collectors and Their Applications
tions in Figs. 11.3.1-11.3.7 vanish.) Equating the instantaneous peak output of the collector field of area A = AAamp with load rate L, we find
Before solving for Adamp we note that F-m has an implicit dependence on collector area; since the overall flow rate m is fixed, the flow rate per collector area and, by Eq. (12.3.3), Fm will change when the collector area is changed. For the following discussion it is convenient to recast this equation in the form
After inserting Eq. (12.3.3) for Fm in Eq. (12.3.8), one obtains
which can be solved for /4 dump . For direct systems (Fx = 1) the result can be written as
For systems with heat exchangers the result is slightly more complicated because Fv also depends on AAump. Inserting Eq. (10.4.11) for Fv one obtains
Since the economically optimal area is likely to be fairly close to Adumi,, it is instructive to calculate the corresponding yearly useful solar energy Q explicitly. Combining Eqs. (12.3.5) and (12.3.8) one can write the yearly useful energy collected with a system of area A = ^dump in the form
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Sometimes the performance of a solar energy system is expressed as solar fraction / denned in the ratio of useful solar energy to load, in our case
where rioad is the number of seconds per year during which there is a load. Equation (12.3.12) gives an explicit expression for the solar fraction for A
From Figs. 11.3.1-11.3.7 we see that q(X) is in the range of 4-8 GJ/(m2 y) for most climates. Typical thresholds are in the range of 0.0-0.3 kW/m2. Taking 0.8 kW/m 2 as a typical value of (7max — X), one finds
for loads of 24 h/day in clear climates. In cloudy climates the solar fraction /at A = ^ dump is smaller. Now consider what happens if A is made larger than Adump while the total flow rate m is kept fixed. In that case the collector supplies more than the load whenever the insolation exceeds the threshold Xdump:
which follows from Eq. (12.3.8) (with Xdump instead of 7max). The annual discarded energy Qdump is equal to the energy that could be collected by a system with threshold ^dump. Thus it can be calculated by Eq. (12.3.5) if one simply replaces X by ^dump:
The useful energy delivered by the system is the difference between the energy Q(X) that could be collected in the absence of dumping, and the discarded energy Qdump:
Inserting Eqs. (12.3.15) and (11.3.2) we can write Eq. (12.3.17) in the form
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Active Solar Collectors and Their Applications
Before we can optimize, we must display all dependences on collector area A explicitly. The only area dependent quantity in Eq. (12.3.18) is Xdump, for which we substitute Eq. (12.3.15) to obtain
Recalling Eq. (12.3.3) for Fm, we obtain
This is the final result when A > Adump. In addition to the coefficients q\ and
on process load rate L, on the flow rate-specific heat product me, and on the collector parameters Fmt)0 and Fm U. EXAMPLE 12.3.1
For a textile factory water must be heated from the water mains temperature to a process temperature 7ioad = 80°C. The water mains temperature is equal to the average ambient temperature Tamb = 19°C. The flow rate is constant at m = 0.75 kg/sec for 24 h/day but the plant operates only 290 days of the year, the rest being downtime due to weekends, holidays, and maintenance. The latitude is 32°N, the yearly average daytime beam irradiance 7h = 0.523 kW/m2, and the peak irradiance on a flat plate collector at tilt equal latitude is /max = 1-0 kW/m2. Consider the use of a flat plate collector array (including losses from pipes) with the parameters
as preheater, without heat exchanger and without storage. SOLUTION
Since the load is year-round, the flat plate should be deployed at tilt equal latitude. For this location the correlation of Fig. 11.3.1 yields
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The threshold X is zero in this case because Tin = T3mb. Following Section 4.2 we find for this flow rate Fin?70 = 0.574. The load rate is
The threshold area for dumping is, by Eq. (12.3.10),
and with this area the yearly useful energy is
Equation (12.3.14) shows that such a system supplies/= 23.4% of the 24h load. It is of interest to note that the annual average system efficiency is KFinr)0 = 0.92 X 0.574 = 53% (apart from the factor 290/365 = 0.79 for holidays and weekends). This is significantly higher than the efficiency of typical residential solar systems. If the area is increased beyond the dumping threshold, Q is calculated from Eq. (12.3.20), again with the correction 290/365 for actual plant operation,
12.3.3 Single-pass open-loop system with storage Many industrial processes consume hot water as opposed to reusing it. This is particularly common in the food and textile industries. For example, in the beverage industry the water used for washing returnable bottles is so contaminated that it must be discarded. One may be able to recover some of the energy with heat exchangers to preheat the incoming fresh water, but the fluid itself is not reused. This feature is of crucial importance for the design of solar heat systems with storage. As shown in this subsection, one can design single-pass open-loop systems for this type of application that are significantly more efficient and economical than conventional multipass closed-loop systems. This single-pass strategy can also be applied in closedloop systems, even high temperature oil systems, but there it requires an additional tank to hold one day's worth of process fluid. The flow diagram for the single-pass open-loop system with storage is shown in Fig. 12.3.2. Figure 12.3.2a shows a direct system, Fig. 12.3.2b an
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Active Solar Collectors and Their Applications
Figure 12.3.2 Single-pass open-loop waterbearing design with variable volume tank, (a) Direct, (b) With heat exchanger (From Collares-Pereira et al. [1984]).
indirect system where a heat exchanger separates the collector from storage and process. The heat exchanger may be necessary either to circumvent the freezing of water in the collectors (in cold climates) or in cases where possible corrosion of collectors by water is to be avoided. Our analysis covers both systems, the only difference being an additional heat exchanger factor Fx for the indirect system. For simplicity we consider only the case of greatest practical importance; namely, a process that is constant in temperature and flow rate. Fluid from the water mains is passed through the collectors at a constant flow rate m whenever the collectors can deliver useful heal. Water enters the collectors at temperature T{n. The outlet temperature 7"out varies with insolation. The heated water flows into the storage tank. Whenever hot water is available in the tank, it is withdrawn to the process at constant flow rate m,oad. The tank should be mixed because that ensures best storage utilization in the single-pass open-loop design. The backup heater is in series with the solar system and operates at variable heat rate to bring the fluid up to the process temperature T'k)ad. When no solar heat is available, the incoming
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347
water bypasses the solar system and goes directly to the backup heater. The storage tank starts out empty at collector turn-on time and is sized to fill up with solar heated water during the course of an average sunny day. The storage tank is a variable volume tank. It need not be pressurized because it is not exposed to the pressure of the water mains (typically on the order of 5 times atmospheric pressure). Instead, the closed tank can have an internal insulating float. Such tanks are commercially available at much lower cost than pressurized tanks, but only for use below the boiling point. This system design stands in marked contrast to the conventional multipass closed-loop design of Section 12.3.4. In the latter the fluid is returned from storage to the collector for further heating. The flow rate is so high that the storage volume makes on the order of 5-10 passes through the collector during a single sunny day and the temperature rise for each pass is small. In a closed-loop water system the tank is exposed to the water mains and must be pressurized. According to the conventional wisdom, the low flow rates of the singlepass design imply a low collector efficiency. It is certainly true that for a given inlet temperature the outlet temperature goes up as the flow rate is decreased and hence the instantaneous collector efficiency decreases. However, in a multipass system the inlet temperature rises during the course of the day and hence the efficiency decreases anyway. In a single-pass system, collection efficiency decreases across the collector from inlet to outlet while in a multipass system it decreases during the course of the day. On balance the daily average collector efficiency is essentially the same—if not slightly higher for the one-pass system—as shown by a theorem of Gordon and Zarmi [1981] and by the computer simulations of Mertol et al. [1981]. The system efficiency of the single-pass system actually turns out to be significantly higher than for the multipass design, as demonstrated by the examples in this chapter. The lower system efficiency of multipass closedloop systems is primarily caused by mixing in storage, during charging and/ or during discharging, an effect that prevents complete utilization of the heat in the storage tank, quite apart from the larger pumping energy. To explain the selection of storage mass M and collector flow rate m let us for the moment assume that all days have the same operating period rop, given by Eq. (11.3.34). The length of time per day when the collector does not operate is rday — rop with rday = 24 h = 86,400 sec. During this time, any available hot water is withdrawn from storage at the process flow rate mload. Clearly one should not make storage any larger than mload(Tday — rop) because any storage in excess of this value would never be depleted; i.e., it would be useless. As shown by Collares-Pereira et al [1984] this storage size is actually close to the optimum; hence we shall assume a storage mass
To determine the flow rate m through the collector we note that the collector is to supply hot water both to the process, at rate mload and to storage.
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Active Solar Collectors and Their Applications
If the tank is to just fill up during the operating period, then the net inflow m — rn\0aa into the tank must be equal to Af/rop. This fixes the collector flow rate in terms of storage mass M as
If the flow rate were larger, hot water would be discarded at the end of the day, while smaller flow rates would leave the top of the tank unused. If the collector area is A, then the annual collected solar energy is, by Eq. (12.3.5),
The net useful energy Q delivered by the system during the year is obtained by subtracting losses from storage <2loss and any excess energy QAumv (if any) that cannot be utilized,
In well designed systems Qloss is a few percent of Q; hence it need not be calculated very accurately. For example, if Q]oss is 5% of Q, then a 20% error in doss causes only a 1% error in Q, such a small error is insignificant. Hence we estimate <2ioss by considering the steady state heat loss from the storage tank. The average tank temperature at the end of an average day's energy collection rslor can be obtained from the daily energy balance of the tank. Since the daily water flow through the collectors is mrop the water mass M collected in the tank by the end of the day represents a fraction M/(rhTop) of the total water flow. All the water from the collectors passes through the tank, where it is mixed. Therefore the fraction M/(rarop) of the water flow approximately equals the fraction of the daily collected energy Gx>n/365 that is in the storage tank at the end of the day. Thus one obtains the energy balance of the storage tank as
where c = 4186 J/(kg K) is the specific heat of water, (2coll is in GJ/yr, Mis in kg, rop is in sec/day, and m is in kg/sec. _ Using Eq. (12.3.23), we readily solve for Tstm. If the tank is located in an environment at average temperature Tenv then the annual loss from storage is approximated by
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where (UA) slm is the effective tank surface-heat transfer coefficient of the tank [in W/°C]. To complete the calculation of Q we need to know the excess energy Qdump. If the collector area is sufficiently small in relation to the load then no energy dumping will ever occur and the calculation is complete. Before worrying about energy dumping, it is advisable to check how large the collector area can be without exceeding the upper tolerance temperature of collector or heat transfer fluid during periods of peak insolation. Typically one wants to keep the collector outlet temperature below 90-95°C to prevent boiling in the collector. Suppose the peak outlet temperature is specified as Tmax, corresponding to peak insolation 7max. Then the largest permissible collector area Amm is given by the instantaneous energy balance
where Fin is evaluated at A = Amm. Inserting Eq. (12.3.3) for F-m and solving for Am^ one finds
In the single-pass open-loop design with fixed flow rate there are two distinct mechanisms that can cause energy dumping: mass dumping and temperature dumping. Mass dumping occurs if a larger mass of heated water is collected than can be consumed by the process. Our choice of storage mass, Eq. (12.3.21), and flow rate, Eq. (12.3.22), guarantees that mass dumping does not occur, at least if all days have the same operating period. CollaresPereira et al. [1984] find that the effect of mass dumping is negligible even when variations in operating period are taken into account. Temperature dumping occurs if the storage tank temperature exceeds the process temperature. One could minimize temperature dumping by adding a tempering valve and varying the flow from storage to process; this complicates the controls a little. Furthermore the use of a tempering valve will not necessarily solve the problem due to conservation of mass, wherein the heated water not consumed on one day by the load will lead to mass dumping the following day. We opt instead for a choice of collector area that avoids temperature dumping. As a simple criterion, for avoiding temperature dumping, we take the condition
where Adump is the collector area that corresponds to the onset of temperature dumping, and a is a number between zero and unity. As shown by CollaresPereira et al. [1984], proper solution of the problem of temperature dumping
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Active Solar Collectors and Their Applications
shows that selecting
affords a conservative estimate of Adump; i.e., a lower limit for Adump. Since F-m depends on area, this is not yet the explicit solution for Adump. After inserting Eq. (12.3.3) for F-m with A = Adump, one obtains the explicit result
If A is less than this value then we can safely assume that energy dumping is negligible. Note that energy dumping for a system with storage involves the daily, rather than the instantaneous energy balance. For that reason ^dump for the storage system is different from Eq. (12.3.10). EXAMPLE 12.3.2
Consider an industrial hot water load in Sde Bpker, Israel (latitude = 30.9°N, yearly average daytime beam irradiance 7b = 0.502 kW/m2, and average ambient 2"amb = 18.3°C). The process runs 365 days/yr, 24 h/day, and consumes water from the mains (at a temperature equal to Tamb), heated to a process temperature Tload = 70°C. The flow rate is wload = 1.0 kg/sec. Design a single-pass open-loop system with storage, using a flat plate array with A = 970 m2 and efficiency parameters Fmi)0 = 0.75 and FmU = 5.0 W/ m 2 °C. The storage tank has height = diameter and heat loss coefficient f/stor = 0.5 W/m 2 °C. Calculate the yearly useful solar energy. SOLUTION
From Fig. 11.3.1 we find the correlation
The threshold is X = 0 because 7'in = ramb. Then Eq. (11.3.34) yields the average operating time
With that the storage mass is chosen according to Eq. (12.3.21) as
The collector flow rate is, from Eq. (12.3.22),
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Applications
Now we transform from Fm to Fin according to Section 4.2, with the result
Before proceeding to the calculation of Q, let us check the upper limits on collector area imposed by temperature limits and energy dumping. With a maximum outlet temperature of 95°C to prevent boiling in the collector, Eq. (12.3.28) yields an upper limit Am^ = 1338 m2. The onset of energy dumping occurs at an area AAump = 1058 m2, by Eq. (12.3.31). Hence we can safely assume that Q dump = 0. The yearly collected energy is, from Eq. (12.3.23),
To find the tank losses we estimate the mean storage temperature from Eq. (12.3.25) (with T i n = ?;mb)as
The tank has radius 1.99 m and surface Aslor = 74.9 m2; hence (UA)slor = 37.5 W/°C. With Eq. (12.3.26) we find an annual tank loss Qloss = 37 GJ, less than 1% of QmH. Thus the final result is, from Eq. (12.3.24),
and the solar fraction is/ = 0.60 of the 24-h load. 12.3.4 Closed-loop configurations If the process fluid is not discarded, one can still employ the basic ideas of the single-pass open-loop design, provided one adds a second storage tank to hold one day's worth of return fluid from the process. This is shown in Fig. 12.3.3. The analysis is the same as for the system of Fig. 12.3.2.
Figure 12.3.3 Single-pass design with two variable volume tanks (From KLutscher et al. [1982]).
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Active Solar Collectors and Their Applications
With a single storage tank there are two different system configurations, as shown in Figs. 12.3.4 and 12.3.5. For simplicity they are only shown as direct systems, but a heat exchanger can be included between the collector and the rest of the system, just as in the other cases that we have considered. The configuration of Fig. 12.3.4 seems to be the most common. It has usually been the preferred choice in residential applications. This system may be called "multipass" design because it is operated at such high flow rates that the storage volume makes on the order of 5-10 passes through the collector during the course of a single sunny day. The tank has four pipe connections: one inlet-outlet pair for the collector loop and one inlet-outlet pair for the storage loop. This arrangement permits the two loops to operate independently of each other. Thus the collectors can provide heat to storage whether or not the load is withdrawing heat from it. Conversely the load can be supplied from storage regardless of whether the collectors are running. This flexibility is desirable in applications where the load is highly variable, as it is in space heating and domestic hot water. There is, however, a performance penalty associated with this design. Since this system is operated at high flow rates, the tank tends to be more or less mixed. Consequently the collector temperature is higher than the load return temperature. In space heating this penalty need not be too severe because the temperature difference between load supply and load return is usually small. For industrial process heat the multipass design is generally not the best choice. The temperature difference between load supply and load return is usually quite large, and the performance penalty of the multipass design can be severe. Also there is no need for independent operation of the collector and of the load loop, because the load is constant. Thus one can use designs with significantly higher efficiency, either the single-pass open-loop design of Fig. 12.3.2 or the two-pipe storage configuration of Fig. 12.3.5. This configuration has been called "two-pipe configuration" by Kutscher et al. [1982] because only two pipes are connected to the storage tank, in contrast to the four pipes in Fig. 12.3.4. If the flow rate is chosen so that the storage volume makes one pass through the collector on a sunny day and if the tank remains
Figure 12.3.4 Multipass closed-loop design (From Kutscher ctal. [1982]).
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Figure 12.3.5 Two-pipe storage configuration. (From Kutscher et al. [1982]).
stratified (as it should since the flow is slow), the performance is essentially equal to that of the single-pass open-loop design. As for the performance analysis, the multipass design can be treated with the 0,/chart method of Section 11.5. Since the multipass design has been so popular, we present a numerical example that quantifies the performance penalty relative to a single-pass design. EXAMPLE 12.3.3
Consider the same application as in Example 12.3.2, but designed according to the multipass configuration of Fig. 12.3.4 with a flow rate per collector area rh/A = 0.015 kg/sec m2 (as recommended for this design in Table 12.1.1), about 6 times larger than in Example 12.3.2. Take the same collector area A = 970 m2 and the standard storage capacity of 75 kg water per m2 of collector. (Note that this implies a total storage mass of 72,750 kg, much larger than the 49,800 kg employed in the single-pass design of Example 12.3.2.) The collector is the same as in Example 12.3.2; it has parameters F,,Mo — 0.75, FmV = 5.0 W/m2 °C and average incidence angle modifier K = 0.92. SOLUTION
As pointed out by Pearson, Klein, and Duffle [1981], the performance of this system type can be calculated not only by the general $, /chart method, but if the system parameters are in the range of the original /chart method one may use the /chart for water heating. In this example the parameters (in particular, the storage size and the water temperature) are indeed compatible with the/chart and hence we use the/chart because it is simpler. The monthly results are listed below in tabular form. Hh is the daily average hemispherical solar irradiation on the horizontal. The daily average irradiation H on the aperture is calculated according to Eq. (11.4.1) with cutoff time tc = ts. The parameters A} and Yf of the /chart correlation are obtained from Eqs. (12.2.1) and (12.1.2), respectively. Adding up the monthly sola energy one finds a solar fraction / = 0.48 for the year.
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Active Solar Collectors and Their Applications
Jan. Feb. Mar. Apr.
May Jun. Jul. Aug. Sep. Oct. Nov. Dec.
7Iamb fC)
77h (MJ/m 2 )
H (MJ/m 2 )
9.6 11.6 13.6 17.0 20.9 24.2 24.8 25.3 23.6 20.6 16.0 11.4
11.9
16.8 18.3 19.4 21.9 23.1 25.0 24.8
14.4 17.5 22.0 25.3 28.8 28.0 25.9 22.1 17.8 13.9 10.3
Yearly/' 0.479
24.7
23.5 21.6 19.1 14.9
xf 3.18 3.08 2.98 2.80 2.60 2.43 2.40 2.37 2.46 2.61 2.85 3.09
Y
f 0.60 0.65 0.69 0.78 0.82 0.89 0.89 0.88 0.84 0.77 0.68 0.53
./load
/
(GJ per mo)
0.346 0.391 0.426 0.499 0.537 0.593 0.589 0.589 0.555 0.500 0.425 0.298
201 205 248 280 312 333 342 342 312 291 239 173
Yearly solar energy [GJ] 3277.47
By comparison the single-pass open-loop design of Example 12.3.2 yielded an annual solar fraction of 0.60, which is 25% higher even though it uses less storage; for the same storage, the difference in solar fraction would be even larger. One key reason for the thermal superiority of the one-pass open-loop design is the fact that all energy stored on a given day is depleted by the nighttime load by collector turn-on time the following day. The more conventional system design, with mixed storage, can never fully deplete the energy stored on a given day by collector turn-on time the following day. The single-pass open-loop design has an additional advantage that does not show up in a comparison with/chart or 0,/chart results because transients are neglected. In a single-pass design penalties due to collector warmup and cool-down are much smaller than in a multipass design because the collector operates at lower inlet temperature and any heat above Tm is useful since the solar system acts as preheater only. The low flow rate has another benefit beyond the comparison of thermal efficiency. In multipass systems the parasitic power consumption for pumping has been found to represent typically 3%-10% of the collected energy. Since pumping energy is supplied as electricity, its cost can easily eat up a sizable portion of the energy savings from solar heat. 12.4 CENTRAL HEAT COLLECTION In large solar process heat installations and solar thermal power plants the transport of heat to a central point of use raises some problems. Basically there are two approaches: either an array of distributed thermal collectors, from which heat is brought to a central point by means of a piping network; or else the optical transport of energy in a central receiver. Choosing the best approach for a given application may involve complicated trade-offs between collector costs, piping costs, collector spacing, optical losses due to
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Applications
shading, heat losses, etc. For example, by placing the collectors very close together one minimizes piping costs and piping losses, but at the price of poor optical performance due to shading. There are, however, some simple general recommendations that have emerged from many detailed studies. The higher the temperature, the higher the cost and the losses of a heat collection network. This gives the central receiver approach a clear advantage at high temperatures. To be more quantitative we summarize the results of a study by lannucci [1980]. In this work, parabolic trough, parabolic dish, and central receiver systems were compared for industrial process heat applications at three temperatures: 93°C, 149°C, and 316°C. System size ranged from 3-1500 MW, for each. Each system was optimized to deliver heat at the lowest possible cost. The results are summarized in Tables 12.4.1 and 12.4.2. Table 12.4.1 gives the cost of the heat centralization network, in $/m2 of collector aperture. A range of values is shown, corresponding to the range of system sizes. Generally the cost increases with temperature. The most striking result is the difference between the three collector types. For the central receiver the heat centralization adds only 0.5-2.5 $/m2 of aperture, while for the dish system it adds 50-100 $/m2. The latter is quite large, considering that total system costs should remain below about 200 $/m2 to be competitive with fossil fuel. The trough system falls in between the dish and the central receiver in this comparison. The piping heat loss results of Table 12.4.2 show a similar trend. For the central receiver they are very small, a fraction of a percent of the collected energy. For the dish, on the other hand, they are in the range of 4% to 20%, increasing with temperature. Again the trough system is intermediate in its losses. The reason for these results is easy to understand from the length of the piping network required in each case. For example, the 1500-MW, system requires roughly 6, 130, and 1000 km of pipe length for the central receiver, the trough, and the dish system, respectively. The central receiver only needs a riser and a downcomer between top and base of the tower. A trough system needs less piping than a dish system because the troughs can be interconnected in a way that requires little additional piping; the troughs themselves are part of the centralization network. In a dish system, on the other TABLE 12.4.1 Thermal Energy Centralization Cost Summary (in $/m2)a Technology
temperature (°C)
Central receiver
Trough
Dish
93 149 316
0.5-1.5 0.5-1.0 1.5-2.5
7-7.5 7-8 10-16
50-55 51-56 69-99
a From lannucci [1980]. All costs in rounded 1979 dollars; the ranges represent variations in system power rating from 3-1500 MW,,,.
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Active Solar Collectors and Their Applications TABLE 12.4.2 Thermal Losses of Heat Centralization Network, as Percentage of Total Delivered Heata Technology
Temperature (•C)
Central receiver
Trough
Dish
93 149 316
0.05- .15 0.10-0.2 0.30-0.7
0.7-0.9 1.2-1.6 3.6-3.8
3.6- 4.8 5.0- 7.0 16.6-20.4
"From lannucci [1980].
hand, heat is focused at individual points and all these points must be connected by pipes. Of course, the heat centralization network is only one part of the system, and what really matters is the total cost per energy. To some extent the high heat centralization penalties of the dish system are compensated by superior optical performance. As shown by Fig. 3.8.1, the dish, being always normal to the sun, receives about one-third more radiation than the trough. The radiation incident on the heliostats of a central receiver depends on the design details, but generally it is comparable to or even a little less than the radiation incident on parabolic troughs with east-west axis. Which of the three collector types is best for a given process heat application must be decided by weighing the energy centralization penalties of Tables 12.4.1 and 12.4.2 against the cost and energy collection capabilities of the different collectors. In the temperature range up to about 300°C there does not appear to be a clear general favorite at the present time.2 However, for power generation the case is different. While many industrial applications can use heat below 300°C, for power generation one needs far higher temperatures, 500-1000°C, to obtain acceptable system efficiency. The heat centralization penalties in Tables 12.4.1 and 12.4.2 increase so rapidly with temperature that a solar thermal power plant with distributed collectors and central conversion does not appear to be practical. Hence for high-temperature solar thermal power conversion there appear to be only two practical solutions: the central receiver and the point-focus-distributed collector (e.g., dish), with heat engines at the focus of each collector and collection of energy as electricity. 12.5
POWER GENERATION
There are many ways of converting solar radiation to mechanical or electrical power. Some of these (e.g., wind or wave power) involve natural conversion processes. Since this book is concerned with active solar collectors, 2 The comparison between dish, trough, and central receiver looks different if the load is small enough to be satisfied by a single dish or at most a few dishes. In that case the cost of collecting heat is about the same for the dish as for the trough system, and the size may not be large enough to make a central receiver practical. Thus the dish may turn out to be the preferred collector for small high-temperature process heat loads.
357
Applications
we address only photovoltaic and solar thermal conversion schemes. The major options are listed in Table 12.5.1 and discussed below. 12.5.1 Photovoltaics The field of photovoltaics is in a state of flux (e.g., see Sunworld, 1980, 1982; Backus, 1980; Carlson, 1982; Wolf, 1980]. Vigorous research and development programs are in progress in several countries and are producing many new results. There are certainly many unexplored possibilities since the photovoltaic effect can take place in a large number of semiconductor materials and material combinations. At the present time, single crystal silicon cells are the best known and the most developed. But other materials, especially amorphous silicon, cadmium sulfide (CdS), cuprous sulfide (Cu2S), and gallium arsenide (GaAs) may hold more promise for the future. There are two basic approaches towards reducing the cost of photovoltaic electricity, as outlined in Table 12.5.la. One can try to reduce the cost of the cells and deploy them as simple flat plates. Alternatively one can use concentrators to reduce the required area of the cells. Many combinations of cells and concentrator optics have been studied and the consensus has emerged that in the long run the extremes of flat plate and high concentration (C > 100) hold the greatest promise [Bos and deMeo, 1978]. Commercial viability also seems to require that the cell efficiencies be above certain lower bounds, on the order of 10% for flat panels and 25% for cells in concentrators. With lower efficiencies the costs of the nonphotovoltaic system components (land and support structures) would become prohibitive. Currently the efficiency of thin film cells (e.g., amorphous silicon) is around 7%10% for laboratory samples and lower in mass production, but with further TABLE 12.5.la Photovoltaic Power Generation Schemes Cell efficiency [%] Collector type Flat plate
Concentrator
Cell type
Typical
Highest demonstrated
Single crystal Si Polycrystalline Si Amorphous Si, CdS, Cu2S
10-14
18
Best developed
4-6 5-8
10 10
Low-cost thin film cells, probably cell efficiency above 10% needed for commercial success
Single-crystal Si GaAs
15
20 26
High cell efficiency (probably 25%) needed for commercial success; probably best with point focus and high concentration; utilization of waste heat may be practical
Comments
358
Active Solar Collectors and Their Applications
research and development one expects to surpass the 10% hurdle. Cell efficiencies about 25% can be reached with GaAs and with combinations of cells that are matched to different portions of the solar spectrum. From the point of view of systems analysis one needs to understand how the cell output depends on operating conditions. The principal determining factors are cell temperature and insolation level. By contrast to thermal collectors the variation of photovoltaic efficiency with insolation is quite weak: the efficiency increases only logarithmically with insolation.3 For systems calculations one usually assumes a single constant efficiency value corresponding to intermediate insolation levels. The variation of cell efficiency with temperature is quite pronounced and can be approximated by a linear relationship. For a typical silicon cell the temperature variation is characterized by
The temperature variation of cell efficiency for several cell materials can be estimated from Fig. 12.5.1. The dashed curves indicate the theoretical maximum cell efficiency as a function of energy gap for various temperatures. The solid lines show the energy gaps of the most important cell materials. For example, one sees that the maximal efficiency of Si cells decreases from 24% at 273 K to 15% at 373 K. The temperature effect makes it necessary to pay attention to cell cooling. For flat panels cooling by natural convection of ambient air will usually be 3 This logarithmic variation allows the attainment of higher cell efficiencies with concentrators. A cell efficiency of 20% has been demonstrated with a specially designed silicon cell under high concentration [O'Donnell et al., 1978].
Figure 12.5.1 Maximum efficiency as a function of energy gap Ex for various temperatures (From Wolf [1980]).
359
Applications
adequate; but for concentrators, active cooling may be required. If there is demand for low-temperature heat, the heat from the cells can be collected and utilized. This cogeneration of heat and electricity can be accomplished both with flat panels and with concentrators. Concentrators are better suited for cogeneration because the heat is already concentrated and the incremental cost of the necessary heat collection equipment is low. 12.5.2 Solar thermal power The foremost constraint on solar thermal power generation is imposed by the Carnot efficiency
where rhigh and Tiow are the absolute temperatures of heat source and sink, respectively. This equation can be taken as guideline for systems considerations because in practice most heat engines can achieve roughly half of the Carnot efficiency.4 Therefore there are two extreme approaches to solar thermal power generation: low-temperature schemes with a cheap or free source of heat, and high-temperature schemes with relatively expensive collectors but high efficiency. Between these extremes lies a spectrum of possibilities as sketched in Table 12.5.1b. 4 At very low-temperature differences the efficiency achievable in practice tends to be lower than this rule of thumb because of the difficulty of transferring heat across very small temperature differences.
TABLE 12.5. Ib Solar Thermal Power Generation Schemes Collector type Ocean Solar ponds
Flat plates, evacuated tubes, collectors with low or intermediate concentration (C < 50) Point-focus high concentration central receiver
Point-focus parabolic dish Fresnel lens
Conversion scheme
Comments
Closed Rankine cycle (ammonia or freon) or open cycle with water Organic Rankine cycle
Very low efficiency but free heat; only practical for baseload power Low system efficiency (—1%) but practical for sunny regions with salt lakes; built-in storage (a few weeks to a few months)
Organic Rankine cycle
Fairly low efficiency; collectors must deliver heat at a fraction of the cost of heat from fossil fuel
Steam Rankine cycle Gas turbine (Brayton cycle)
10 MWe pilot plant operating; utilization of waste heat may be practical (cogeneration)
Stirling cycle
Under development
360
Active Solar Collectors and Their Applications
The surface waters of the tropical oceans reach temperatures of 25-30°C while the bottom, even in the tropics, remains at 4°C. This represents an enormous amount of free and renewable heat that can be converted to electricity. This electricity can be transmitted to shore or used by energy intensive industries on floating platforms. An ocean thermal power plant is under test off the shore of Hawaii. Salt-gradient solar ponds are a source of low cost heat up to the boiling point of water. The system efficiency of a solar pond power plant is on the order of 1 %. This is acceptable if solar ponds can be built at extremely low cost, something that should be possible with salt lakes in sunny regions. A demonstration solar pond power plant (with 150 kWf peak and about 20 kWP average power) is being operated at the Dead Sea in Israel. Solar pond and ocean thermal power plants are unique among solar power plants because they can supply baseload electricity. When solar thermal power plants received widespread attention in the early seventies, many systems were proposed based on organic Rankine cycles powered by flat plates, evacuated tubes, parabolic troughs, or other low- and intermediate-temperature solar collectors. This made sense as long as the cost of point-focus high-temperature collectors was expected to be much higher than that of collectors for lower temperatures. But in the meantime it has become clear that low- and intermediate-temperature collectors will not be sufficiently cheaper than high-temperature point-focus collectors and the latter will be preferable for power generation by virtue of the higher conversion efficiency. The examples in Table 12.5.2 illustrate this point by comparing the use of several collector types for process heat and for power generation: a low-temperature system (r high = 90°C) with flat plate collectors, an intermediate temperature system (rhigh = 300°C), and a high-temperature system (7i,igh = 600°C) with central receiver.5 The first line lists system cost estimates C in $/m2 of aperture area, assuming mass production. The next few lines list thermal efficiency and delivered heat q,, in GJ,/ m 2 of aperture, for a sunny location at midlatitudes. Needless to say, there is some arbitrariness in the choice and performance figures since both are estimates of realistic goals for future technologies. However, the readers will find that the basic conclusion about the ranking of different power options remains unchanged over a wide range of input assumptions. As discussed in Chapter 14 the cost of solar heat is the ratio of system cost C per aperture area [in $/m2] and delivered heat #, multiplied by the annual charge rate ~fa\
For the annual charge rate we take 0.10 in constant currency as a typical value. Similarly the cost of solar electricity is JaC/qt^ where qe is the annual 5
As a reference on central receivers, see, e.g., ASME [1984].
361
Applications TABLE 12.5.2 Cost and Performance Comparison of Hat Plate, Parabolic Trough, and Central Receiver for Process Heat and for Power Generation
1.
2. 3. 4. 5. 6. 7. 8. 9.
System cost, per aperture area ($/m2) Temperature 7"high (°C) Yearly irradiation on aperture (GJ/m2 yr) Yearly average thermal efficiency (%) Annual collected heat per aperture area (GJ,/m2 yr) Cost of heat ($/GJ,) if annual charge rate /cr = 0.10 in Eq. (12.5.3) Electric conversion efficiency (%) taken as lcW2 with rlow = 293 K(20°C) Annual electricity production per aperture area (GJf/m2) Cost of electricity ($/GJe) if annual charge rate /cr =
0.10
Flat plate
Parabolic trough
Central receiver
150 90 8.0 30%
170 300 6.8 50%
190 600 6.5 60%
2.4
3.5
4.0
6.3
4.9
4.8
10%
24%
33%
0.23 64.8
0.86 19.9
1.3 14.3
electricity production, in GJ<,/m2 of aperture area. qe is the product of the heat q, and of the electric efficiency ??<, taken as half of the Carnot efficiency. The results show that while all three collectors can deliver heat at comparable costs [4.8-6.3 $/GJ,], the high electric conversion efficiency puts the central receiver at a clear advantage when it comes to generating power. While the conversion efficiency from heat to electricity in a point-focus collector (33%) is comparable to the efficiency of a fossil-fired power plant, the efficiency of low-temperature solar thermal schemes is very much lower. Thus the latter approach to power generation can be practical only if lowtemperature collectors can deliver heat at a small fraction of the cost of heat from fuel. This is very unlikely. Certainly the flat plate system has such a low electric efficiency as to be uneconomical for any reasonable collector cost. Even for the intermediate-temperature scheme the electric efficiency is fairly low, and hence the parabolic trough appears to be a poor candidate for power generation. REFERENCES Anderson, B. N. and Michal, C. J. 1980. "Passive Solar Design." In Chapter 31, Solar Energy Technology Handbook, Dickinson, W. C. and Cheremisinoff, P. N., editors. New York: Marcel-Dekker. ASME. 1984. The February 1984 issue of the ASME J, Solar Energy Eng. 106:2 103 is devoted to central receivers. Backus, C. E. 1980. "Principles of Photovoltaic Conversion." In Solar Energy Technology Handbook, Dickinson, W. C. and Cheremisinoff, P. N., editors. New York: Marcel-Dekker. Balcomb, J. D. 1980. "Passive Solar Energy Systems for Buildings." In Chapter 16 Solar Energy Handbook, Kreider, J. F. and Kreith, F., editors. New York: McGraw-Hill.
362
Active Solar Collectors and Their Applications
Beckman, W. A., Klein, S. A., and Duffie, J. A. 1977. Solar Heating Design by thefchart Method. New York: Wiley-Interscience. Bendt, P. and Soto, R. 1980. "Home Owner's Solar Sizing Workbook." Golden, CO: Solar Energy Research Institute. Bos, P. B. and deMeo, E. A. 1978. "Perspectives on Utility Central Station Photovoltaic Applications." Solar Energy 21:177. Butti, K. and Perlin, J. 1980. The Golden Thread. New York: Van Nostrand-Reinhold; also, 1980. "The History of Terrestrial Uses of Solar Energy." In Chapter 1, Solar .Energy Handbook, Kreider, J. F. and Kreith, F. editors. New York: McGraw-Hill. Carlson, D. E. 1982. Press Conference, 26 February. RCA. Collares-Pereira, M., Gordon, J. M., Rabl, A., and Zarmi, Y. 1984. "Design and Optimization of Solar Industrial Hot Water Systems with Storage." Solar Energy 32:121. Duffie, J. A. and Beckman, W. A. 1980. Solar Engineering of Thermal Processes. New York: John Wiley & Sons. Duffie, J. A., Beckman, W. A., and Mitchell, J. W. 1980. "Solar Cooling." In Chapter 30, Solar Energy Technology Handbook, Dickinson, W. C. and Cheremisinoff, P. N., editors. New York: Marcel-Dekker. Duffie, J. A. and Mitchell, J. W. 1983. "F-Chart: Predictions and Measurements." ASME J. Solar Energy Eng. 105:3. Engebretson, C. D. 1964. "The Use of Solar Energy for Space Heating—M.I.T. Solar House IV." In Proc. of the UN Conference on New Sources of Energy, Vol. 5, p. 159. Faiman, D., Gordon, J. M., and Govaer, D. 1979. "Optimization of a Solar Water Heating System for a Negev Kubbutz." Israel J'. Technol. 17:19. Fanney, A. H. and Lin, S. T. 1979. "Experimental System Performance and Comparison with Computer Prediction for Six Solar Domestic Hot Water Systems." In Proc. of the 1979 International Congress of the International Solar Energy Society, Atlanta, GA. Farrington, R., Murphy, L. M., and Noreen, D. 1980. "An Analysis of Solar Domestic Hot Water Systems from a System Perspective." In Proc. of the 1980 American Section International Solar Energy Society Meeting, Vol. 3.1, p. 162, Phoenix, AZ. Gordon, J. M. and Zarmi, Y. 1981. "Thermosyphon Systems: Single vs. Multi-Pass." Solar Energy 27:441. Gordon, J. M. and Rabl, A. 1982. "Design, Analysis and Optimization of Solar Industrial Process Heat Plants Without Storage." Solar Energy 28:519. Govaer, D. and Zarmi, Y. 1981. "Analytical Evaluation of Direct Solar Heating of Swimming Pools." Solar Energy 27:529. Govaer, D. 1984. "Determining the Solar Heating of Swimming Pools by the Utilizability Method." Solar Energy 32:667. Grunes, H., deWinter, F., and Kittle, L. 1982, "Solar Augmented Gas-Fired Water Heater." Sunworld6(l):16. Hay, H. R. and Yellot, J. I. 1970. "A Naturally Air Conditioned Building." Mech. Eng. 92(1): 19. Harrigan, R. W. 1980. "Total Energy Systems Design." In Chapter 32, Solar Energy Technology Handbook, Dickinson, W. C. and Cheremisinoff, P. N., editors. New York: Marcel-Dekker. lannucci, J. J. 1980. "Thermal Energy Centralization Networks: Design, Cost and Performance for Dish, Trough and Central Receiver Systems." In Proceedings of
Applications
363
the 1980 American Section of International Solar Energy Society Meeting, Vol. 3.1, p. 44, Phoenix, AZ. Karaki, S., Armstrong, P. R., and Bechtel, T. N. 1977. "Evaluation of a Residential Solar Air Heating and Nocturnal Cooling System." Report COO-2868-3. U.S. Department of Energy. Karaki, S., Duff, W. S., and Lof, G. I. G. 1978. "A Performance Comparison Between Air and Liquid Residential Solar Heating Systems." Report COO-2868-4. U.S. Department of Energy. Kirkpatrick, D. K., Masoero, M., Rabl, A., Roedder, C. E., Socolow, R. H., and Taylor, T. B. 1983. "The Ice Pond—Production and Seasonal Storage of Ice for Cooling." Report CEES 149. Center for Energy and Environmental Studies, Princeton University, to be publ. in Solar Energy. Klein, S. A. and Beckman, W. A. 1979. "A General Design Method for Closed-Loop Solar Energy Systems." Solar Energy 22:269. Kreider, J. F. and Kreith, F. 1982. Solar Heating and Cooling, 2nd ed. New York: McGraw-Hill. Kutscher, C. F. 1981. "Design Considerations for Solar Industrial Process Heat Systems." Report SERI/TR-632-783. Golden, CO: Solar Energy Research Institute. Kutscher, C. F., Davenport, R. L., Dougherty, D. A., Gee, R. C., Masterson, P. M., and May, E. K. 1982. "Design Approaches for Solar Industrial Process Heat Systems." Report SERI/TR-253-1356. Golden, CO: Solar Energy Research Institute. Lof, G. O. G. et al. 1964. "Design and Performance of Domestic Heating System Employing Solar Heated Air—The Colorado House." In Proc. of the UN Conference on New Sources of Energy, Vol. 5, p. 185. Mertol, A., Place, W., Webster, T., and Greif, R. 1981. "Detailed Loop Model Analysis of Liquid Solar Thermosyphons with Heat Exchangers." Solar Energy 27:367. O'Donnell, D. T., Robb, S. P., Rule, T. T., Sanderson, R. W., and Backus, C. E. 1978. Arizona State University Report. Pearson, K. A., Klein, S. A., and Duffie, J. A. 1981. "Generalized 0, f-chart Design Method for Solar Hot Water Heating Systems." American Section oflSES Meeting, Philadelphia, PA. Rabl, A. 1981. "Direct Solar Industrial Process Heat." Report CEES 118. Princeton, NJ: Center for Energy and Environmental Studies, Princeton University. Rabl, A. and Nielsen, C. E. 1975. "Solar Ponds for Space Heating." Solar Energy 17:1. Shelpuk, B. 1978. "Proc. of the Desiccant Cooling Conference." Report SERI 22 (1978). Golden, CO: Solar Energy Research Institute. Sillman, S. 1981. "The Trade-off Between Collector Area, Storage Volume, and Building Conservation in Annual Storage Solar Heating Systems." Report SERI/ TR-781-907. Golden, CO: Solar Energy Research Institute. Supple, R. G. 1982. "Evaporative Cooling for Comfort." ASHRAE J. (August):36. Sunworld. 1980, 1982. Special issue on solar photovoltaics, Sunworld4(1) (1980) and 6(3) (1982). Wolf, M. 1980. "Photovoltaic Solar Energy Conversion Systems." In Chapter 24, Solar Energy Handbook, Kreider, J. F. and Kreith, F., editors. New York: McGraw-Hill. Wang, Y. F., Li, Z. L., and Sun, X. L. 1982. "A Once-Through Solar Water Heating System." Solar Energy 29:541. Vitro Labs. 1981. "National Solar Data Program Performance Results." Report SOLAR/0005-81. Silver Spring, MD: Automation Industries, Inc., Vitro Labs Div.
13. PRACTICAL CONSIDERATIONS
Experience is the name everyone gives to their mistakes. Oscar Wilde
The best design is of little use if it is built improperly or with inappropriate materials. The requirements for materials are severe. Collectors are outdoors, exposed to wind, rain, hail, frost, ultraviolet radiation, and dirt. They experience thermal cycling, not only between day and night but in irregular intervals due to clouds. They should perform well over a long time span, preferably at least 10-20 years. And of course the cost should be low. Finally the system must be installed correctly. Even though many components of a solar energy system are familiar from other applications, such as conventional space conditioning or industrial processes, one cannot always rely on experience gained elsewhere. Solar technologies are sufficiently new and unfamiliar that a great deal of learning and experience is required before these technologies can be considered mature. This has been one of the lessons of the solar demonstration projects in the U.S. The sort of problems that were encountered'may be highlighted by that famous story about a flat plate collector imported from Australia to the U.S. The instructions said "Install facing due north," and for once the installers followed all the instructions. .. . The past decade has provided much experience about the practical problems of solar equipment. Engineers and technicians have been trained for solar jobs, and hopefully the errors of the past will be avoided in the future. Much has been written about the practical aspects of solar energy. Within the scope of this book we can present only a brief summary. More can be found in the references at the end of this chapter; in particular, Kutscher [1981] and Kreider and Kreith [1982]. Materials are discussed in Section 13.1, installation problems in Section 13.2, and maintenance in Section 13.3. 13.1
MATERIALS
13.1.1 Absorber
The absorber should have as high an absorptivity for solar radiation as possible, and it should conduct the absorbed heat efficiently into the heat trans364
Practical Considerations
365
fer fluid. Most absorber materials do not have a very high absorptivity, and they need to be covered with special solar coatings. A large number of combinations of coatings and substrates is possible. The absorber substrate must be compatible with the heat transfer fluid, and it must resist corrosion. It must be able to withstand not only the temperature swings of normal operation, but also stagnation conditions (unless some other protection against overheating is provided; e.g., defocusing or shading in case of pump failure). How important the thermal conductivity of the absorber material is depends very much on the design of the collector. In many flat plate collectors the fluid flows through tubes that are bonded to an absorber sheet. Since the tubes are some distance (on the order of 10 cm) apart, good conductivity of the absorber sheet is important. For this design, copper is the preferred material for its high conductivity and corrosion resistance; unfortunately it is also expensive. In typical collectors the fin efficiency of the copper plate is on the order of 95% relative to a plate with infinite conductivity. Aluminum offers fairly high conductivity at a lower cost, but most manufacturers have avoided it because of its susceptibility to corrosion. Of course the bonding between tubes and plate should provide good thermal contact. The conductivity of the absorber material is not critical in designs where the fluid flow reaches the entire absorber area. For example, in a parabolic trough with tubular receiver the heat needs to travel only through the wall of the tube, and with any reasonable material the resistance across the tube wall is small compared to the resistance from the inner tube surface to the bulk of the fluid. (High circumferential conductivity may be desirable to minimize differential heating of tube top and bottom; however, this can also be achieved by making the fluid spiral through the tube.) Similarly the conductivity of the absorber material does not matter in flat plate air collectors, where the air flows past the entire absorber plate. The effect of absorber conductivity on collector efficiency has been illustrated quantitatively by the examples in Section 10.2. The absorptivity a of the absorber coating for solar radiation should be as high as possible. On the other hand one would also like to minimize heat losses by having a low emissivity e for thermal radiation at typical absorber temperatures. Of course KirschhofPs law says that at any one wavelength X, the spectral absorptivity ax must equal the spectral emissivity ex. However, the thermal radiation emitted by solar collectors is contained mostly in the wavelength range above 2 nm while most of the solar spectrum has wavelengths shorter than 2 /urn. This fact allows the possibility of so-called selective absorbers with high absorptance a (integrated over the solar spectrum) and low emittance «(integrated over the thermal spectrum). An ideal selective absorber would have «x - 1 for X < 2 pm and ex = 0 for X > 2 ^ m. The exact value of the ideal cutoff wavelength depends on collector temperature; the value of 2 ^m is approximately optimal. In practice quite a few selective coatings have been developed with « around 0.95 and e around 0.1. The spectral absorptivity for a typical selective coating, black chrome, is
366
Active Solar Collectors and Their Applications
shown by the solid line in Fig. 13.1.1. Nonselective coatings (e.g., ordinary black paints) have a ~ e = 0.95. Some of the most important absorber coatings are listed in Table 13.1.1. For a given solar absorptance a the performance of a collector is always improved if one reduces e. But in practice one does not have the freedom to choose a and e independently; rather, a given coating comes with a given a and t. For some coatings (e.g., black chrome), one can vary a andeby varying the deposition process. Usually an attempt to raise a causes also an increase in e, and careful fine tuning of the coating process is needed to find an optimal coating. The choice of the optimal coating for a given application involves a trade-off between optical and thermal losses.' As a general rule a should be at least 0.85, preferably above 0.90. The performance benefit of selective coatings is greatest in evacuated collectors; in fact if one goes to the trouble of evacuating a collector, then one should also use a selective coating with a low emissivity. In nonevacuated collectors convection begins to dominate if e is reduced below a certain value (0.2-0.5 depending on temperatures and absorber design). In a flat plate collector, for example, the heat loss is reduced dramatically if e is reduced from 0.95 to 0.3, but a further reduction in e from 0.3 to 0.1 brings 'Sometimes the ratio a/e has been used as a ranking index for selective coatings. However, it determines only the stagnation temperature of a surface in vacuum. For practical solar applications the ratio aft by itself is irrelevant.
Figure 13.1.1 Wavelength correlation between solar spectrum, thermal spectrum (100°C And 300°C), and spectral reflectivity of selective surface. Solid line shows black chrome (From Lampert and Washburn [1979]).
TABLE 13.1.1 Properties of Absorber Coatings Material
Type
Supplier/ developer
Absorptance
Emittance
Temperature stability (°C)
Pyromark PbS pigment with silicon binder
Paint Paint
Tempi! Experimental
0.95 0.90
0.85 (500°C) 0.37
750 350
Copper-chromium oxide pigment, ethylene-propylene diene monomer binder
Paint
Harshaw Chemical (pigment) Exxon (binder)
0.92-0.94
0.30-0.45
—
Solkote Hi/Sorb-I
Paint
Solar Energy Corporation
0.95
0.45
870
Black chrome
Electrodeposited
Many
0.94-0.96
0.20-0.25 (300°C)
300
Black nickel (Tabor black) Black copper
Miromit Electrodeposited Chemically Ethone, Inc. deposited
0.91
0.14
—
0.88-0.91
0.12-0.20
190
Comments Expected life less than 2 yr; may require repainting of local hot spots Durability expected to be better than acrylic paint; estimated life less than 5 yr Manufacturer's data for lab application on Al Estimated life greater than 30 yr — Sensitive to humidity
References a b, c
b
a, b
a
b
TABLE 13.1.1 Properties of Absorber Coatings (continued) Material Alcoa black
Type
Absorptance
Emittance
Temperature stability (°C)
0.90
0.30-0.40
175
Seraphin and Carver
0.91
0.11 (at 500°C)
500
University of Sydney
0.92
0.05 (at 120°C)
—
Chemically Alcoa deposited
Chemical vapor deposited Graded metal-carbon Sputtered on copper Silicon nitride over molybdenum
Supplier/ developer
Comments
References
Estimated life greater than 10 yr; coating destroyed by water Withstood 1000 h at 500°C, 1 torr vacuum
b
—
e
d
"Call [1979]; bVresk [1981]; cKreider and Kreith [1982]; dSeraphin and Carver [1979]; 'Harding and Window [1980] and Harding and Moon [1981].
Practical Considerations
369
only a small benefit. Therefore the selective paints with their low cost look especially attractive for flat plates. The optimal choice of an absorber coating depends not only on collector efficiency but also on cost and durability. Generally, paints are cheaper and easier to apply than coatings deposited by chemical or electric processes. As Table 13.1.1 shows, there are quite a few coatings that can be used up to about 200°C, but at high temperatures durability becomes a problem. Black chrome, a favorite because of performance and durability, begins to degrade above 350°C. For temperature stability the Solkote coating looks particularly impressive: it has a = 0.95, e = 0.45, and a temperature limit of 870°C. The higher the operating temperature, the more difficult it is to achieve selectivity. Not only does material degradation become more severe a problem at high temperature, but the emissivity increases with temperature; furthermore, the overlap between solar and thermal radiation spectra becomes greater. Some sophisticated multilayer coatings have been developed that maintain high selectivity up to SOOT; in particular, a coating by Seraphin and Carver [1979] with « = 0.91 and « = 0.11 at 500°C. Above 500°C no coatings with high selectivity are presently available. Many factors influence the durability of absorber coatings. Paints, in particular, are sensitive to ultraviolet radiation and thermal cycling, and they may crack and separate from the absorber substrate. Water vapor and atmospheric contaminants may cause degradation. For example black nickel degrades when exposed to high humidity at high temperature. Freon-blown insulating foams next to the absorber may release hydrofluoric and hydrochloric acid, causing corrosion of the absorber. The absorptivity of a solar absorber can be enhanced by using the cavity effect; i.e., shaping the absorber as a cavity or cavities or texturing its surface. The cavity effect decreases the selectivity of a selective absorber because the small emissivity is enhanced more than the absorptivity which is already large to begin with. A good analysis of the cavity effect can be found in Sparrow and Cess [1978]. 13.1.2 Cover Most collectors have a transparent cover in front of or around the absorber to reduce heat losses. In some collectors multiple glazing is used to reduce heat losses even more. The heat loss could also be reduced by selective coatings, evacuation, or concentration; which approach is most cost effective depends on many factors. In practice one rarely uses more than two covers. The ideal cover is a transparent insulator; it has high transmittance for solar radiation, low transmittance for thermal radiation from the absorber, high durability, sufficient strength and low cost. The properties of the most common glazing materials are listed in Table 13.1.2. Optical losses of the glazing are due to reflection at the glazing surfaces and due to absorption inside the glazing material. The reflection lossdepends on incidence angle and on index of refraction and is given by Fres-
TABLE ! 3.1.2 Properties of Cover Materials3
Material Soda lime float glass Water white low-iron glass Fiberglass reinforced polyester (Sunlite) Acrylic (plexiglass) Polycarbonate (Lexan) Polytetrafluoroethylene (Teflon) Polyvinyl fluoride (Tedlar) Polyester (Mylar) Polyvinylidene fluoride (Kynar) Polyethylene (Marlex)
Index of refraction
Short wave transmittance (0.4-2.5 M)
Long wave transmittance (2.5-40.0 M)
Density (kg/m3)
Thickness (mm)
Maximum temperature (°C)
1.52 1.52 1.54
0.84 0.90 0.87
0.01 0.01 0.076
2500 2473 1399
3.175 3.175 0.635
232 204 NA
5.4 10.6 NA
1.49 1.586 1.343
0.90 0.84 0.96
0.02 0.02 0.256
1189 1199 2148
3.175 3.175 0.0508
93 121 NA
9.0-13.0 24.0-31.0 NA
1.46 1.64 1.413
0.92 0.87 0.93
0.207 0.178 0.230
1379 1394 1770
0.1016 0.1270 0.1016
66 NA NA
0.60 NA NA
1.500
0.92
0.810
910
0.1016
NA
NA
"Adapted from Butler and Claassen [1980] and Jorgenson [1979]. NA = not available.
Cost (in 1979 dollars, $/m2)
Practical Considerations
371
nel's equation, Eqs. (5.2.4) and (5.2.5). Almost all glazing materials for solar collectors have an index of approximately 1.5 and the corresponding reflective loss at normal incidence is 4% per surface or 8% per glazing. Teflon has the lowest index of refraction, 1.34, and the highest solar transmittance, 96%. Absorption losses depend on extinction coefficient and thickness according to Bouguer's law, Eq. (5.2.15). Ordinary window glass (soda lime glass) has high absorption losses because of its iron content. Its overall solar transmittance of 84% for 3 mm thickness is undesirably low. Low iron glass has almost no absorption loss but it is more expensive. Actually it is not so much the iron content itself that matters as the nature of the iron bond since Fe2+ ions absorb much more strongly than Fe3+ ions. Hence one can cut absorption losses by reducing the concentration of Fe2+ ions rather than eliminating iron altogether. For example, Corning 0317 fusion glass has good transmittance despite high iron content [Vitko, 1978]. Many plastics have some absorptivity for the infrared portion of the solar spectrum. Borosilicate glass is excellent; it is stronger than soda lime glass and much more resistant to thermal shock. Its absorption coefficient for solar radiation is very low. Unfortunately it is relatively expensive.2 Two comments must be added to the solar transmittance values in Table 13.1.2. First, these values refer to normal incidence; at larger incidence angles the transmittance is smaller. (See Fig. 5.2.4.) Second, these transmittance values are for total as opposed to specular transmittance. In a flat plate collector only the total transmittance is of interest since the scattered rays also reach the absorber. Even textured glass can be used. On the other hand for a cover of a focusing collector (e.g., an inflated plastic dome enclosure for lightweight heliostats or parabolic dishes) the specular transmittance is relevant, and it may be significantly lower than the values in Table 13.1.2. Surface scratching has little effect on total transmittance but it can seriously reduce the specular transmittance. The infrared transmittance affects radiative heat losses from the absorber. Glass and thick layers of plastic transmit little or no thermal infrared. The thin plastic films are far less effective in blocking infrared. The importance of that point depends on collector design and application, in particular on the emissitivity of the absorber. If the absorber has a low emissivity then radiative losses are small anyway and it does not matter very much whether this radiation goes to cover or to ambient. Table 13.1.2 also lists density, thickness, maximum temperature, and cost of the glazing materials. Of crucial concern is the durability. In this regard glass is unsurpassed, with the possible exception of breakage from thermal shock, hail, and vandalism. But even the risk of breakage can be greatly reduced by tempering [Vresk, 1981]. Glass is hardly affected by ultraviolet radiation and it is very scratch resistant. The durability of glass as collector cover is highlighted by 2 At the present time borosilicate glass is used in the evacuated tubes sold by Owens-Illinois and by Sunmaster, while the evacuated tubes of General Electric are made of soda lime glass.
372
Active Solar Collectors and Their Applications
Jorgensen [1979], who found that samples of 20-40-year-old glass from south facing windows showed less than 2% decrease in transmittance compared to unexposed glass from the same windows. Results of some aging tests are summarized in Table 13.1.3 [Vresk, 1981]. Several cover materials were exposed for 1-3 years in three locations: Chicago, IL, Miami, FL, and Phoenix, AZ. For glass the data indicate slight increase of solar transmittance with time; this is difficult to believe and may be an indication of uncertainties in these measurements. In any case, the durability of glass is excellent, and the transmittance can be maintained at the original value over the lifetime of the collector. The next most durable glazing is acrylic (also known as Plexiglas) provided it contains ultraviolet (UV) stabilizers; i.e., small amounts of UV-absorbing pigments. Lifetimes of acrylic collector covers are estimated to be in the range of 5-10 years. TABLE 13.1.3 Results of Aging Test for Collector Glazing3 Transmittance (%)
Material
Initial
Chicago
Miami
Phoenix
Exposure time (mo)
Self-supporting Regular float glass (% in., Libby-Owens-Ford)
82.6
83.8
84.0
85.1
32
Indefinite
Plexiglass (% in.-thick molded acrylic; Rohm and Mass V-100 type 045)
87.4
86.4
87.7
88.0
35
5-10
Polycarbonate with clear protective coating (Ye in., Mobay "Merlon")
82.8
NA
78.6
73.3
33
Less than 8
87.5
NA
67.3
76.9
32
2-7
990.7
78.8
86.0
88.7
24
2-5
Polyester (3M Scotchpar* 10; 0.002 in.)
86.9
86.0
83.4
NA
12
1.5-2
Vinylidenefluoride homopolymer (0.0038 in., Kynar 450, Penwalt Corp.)
82.5
78.6
81.1
81.6
33
1-5
Value after exposure at
Plastic film Fiberglass-reinforced modified polyester sheet (Kalwall "Sunlite Premium"; 0.04 in.) Thin film plastics Polyvinyl fluoride (0.004 in., DuPont Tedlar)
"From Vresk [1981]. NA = not available.
Estimated life (yr)
Practical Considerations
373
Polycarbonate, also known as Lexan, showed significant deterioration in these tests. The thin film plastics in this table maintained their transmittance reasonably well. Degradation may be aggravated by environmental pollutants, by humidity, and by elevated temperature, thermal cycling, or flapping in the wind. Thin films may have problems with sagging. At the present time there is still much uncertainty about the use of plastic covers in solar collectors. The most reliable plastic, acrylic, offers no cost advantage over glass. The thin films can be significantly cheaper in first cost, but they may have to be replaced after several years. Thus it is not surprising that so far almost all commercial collectors have used glass covers. This situation may well change with further research and development in the chemical industry. This process takes time because of the need for long term outdoor testing of collector materials. But if plastic films with high transmittance and long life are developed, the cost reductions of solar collectors could be very significant [Atkinson and Caesar, 1983]. Another area for progress lies in coatings. Two kinds of coatings are of interest for collector covers: antireflection (AR) coatings and infrared reflecting coatings, also known as heat mirrors. Coatings of tin and/or indium oxide are examples of the latter. They can be added to glass and they have been tested in solar collectors. In their effect on collector performance they are equivalent to selective absorber coatings. The main drawback of the heat mirror/coatings known so far is their low solar transmittance, caused by a high index of refraction. A typical heat mirror with a nonselective absorber will be equivalent to a selective absorber with e around 0.1 and a around 0.8-0.85. Selective absorbers offer better performance at lower cost. This might change as a result of further research; e.g., by combining AR coatings with a heat mirror. The principal advantage of a heat mirror is that the absorber can get very hot without degradation of the selectivity. Antireflection (AR) coatings are produced by adding a layer with an index of refraction intermediate between air and cover material. Ideally the intermediate layer should have an index equal to the square root of the index of the cover, and its thickness should be one quarter wavelength. Several lowcost AR coatings are available for solar collectors. Etching by hydrofluoric acid can produce a gradual transition in the refractive index at the surface of glass. Reflection losses can easily be reduced below 1% per glass/air interface [Peterson and Ramsey, 1975]. This type of etching is microscopically fine and does not increase the sensitivity to dirt or dust [Vresk, 1981]. Thin films of polymers with appropriate index of refraction can also provide lowcost AR coatings. A reduction of reflection losses from 6% to 0.50% at 550 nm has been reported with a coating of this type by Beauchamp [1975]. 13.1.3
Reflector
Reflectors for solar collectors should have as high a reflectivity as possible. Furthermore the reflectors should be highly specular since scattered radiation is likely to miss the absorber. Only collectors with very low concentra-
374
Active Solar Collectors and Their Applications
tion (side reflectors, V-troughs, and CPCs) can use reflectors with relatively low specularity, provided the total reflectance is high. Apart from total internal reflection there are only two practical materials with sufficiently high specular reflectivity to be suitable for solar reflectors: aluminum and silver (Fig. 13.1.2). Uses for total internal reflection in solar collectors are limited; some possible designs have been analyzed by Winston [1976], Rabl [1978], and Smestad and Hamill [1982]. A pure polished aluminum surface has a solar reflectance of 0.90, while polished silver has 0.96. Both silver and aluminum must be protected from the environment; otherwise the reflectance would rapidly deteriorate to unacceptable levels. Unprotected aluminum tarnishes less rapidly because upon contact with air, aluminum immediately develops a thin transparent oxide coating that inhibits further oxidation; however, this coating is not enough to protect against the moisture and pollutants found in most environments. To protect the reflector surface, one places the aluminum or silver layer behind a transparent cover of glass or plastic. This is called a second surface reflector by contrast to a first surface reflector, where the aluminum or silver are directly exposed. The back of a second surface solar reflector must also be protected, but that is relatively easy because one can use opaque coatings. First surface aluminum or silver reflectors can be used in a vacuum, e.g., in the shaped tube CPC collector of Fig. 1.3.6b. The protective oxide layer of an aluminum sheet can be enhanced by an anodizing process to the point where aluminum can be used as first surface reflector outdoors. Aluminum sheets of this type are sold under the tradenames Alzak or Kinglux and are frequently used for lighting and interior decorating. Outdoors they survive quite well in dry unpolluted environments, but in other areas they may tarnish within years or even months. Most reflectors that are exposed to the elements will probably be second
Figure 13.1.2 Spectral dependence of hemispherical reflectivity (From Seraphin and Meinelf 1976]).
Practical Considerations
375
surface reflectors. Glass provides the best protection, but it is rigid and fragile. Most concentrators need curved reflectors, and curving glass can be a problem. One could shape the glass while it is hot, but such a process is very capital intensive and justifiable only by a large production volume. The problems of shaping cold glass mirrors increase with curvature; the stresses induced in the glass make it more susceptible to breakage. In this regard a plastic film like acrylic looks much better. In fact quite a few manufacturers offer solar reflector sheets consisting of aluminized plastic films with adhesive backing. They are easy to handle and to attach to a shaped substrate. The main problem with these plastic reflector foils is durability. The plastic surface scratches easily and loses specular reflectivity. Also, pollutants may reach the metal surface through scratches or pinholes and corrosion will set in. Silver reflectors with plastic coating seem to be particularly vulnerable; so far none have been developed that hold promise of surviving outdoors for 20 years. Absorption in the cover material causes another problem with second surface reflectors. In mirrors with ordinary window glass the absorption is too large to be practical (note that the path of a light ray through the glass is twice the thickness, divided by the cosine of the incidence angle in the glass). Hence one needs either low-iron glass or very thin glass, 1 mm or less in thickness. The latter, also called microsheet, looks most promising for the future. Since microsheet by itself is extremely fragile, one needs to bond it to a thin plastic substrate. Such a sandwich of microsheet glass, silver, and plastic substrates could be rolled and unrolled easily. It combines the highest possible reflectance with durability, and it can readily be bonded to curved reflector substrates. Unfortunately the development costs of this material are high and require a large market. At the present time microsheet reflectors are not yet commercially available, but they do appear to be the solar reflector material of the future. Reflectance and beam spread data for some typical solar reflectors are shown in Table 13.1.4. For the optical analysis of focusing collectors one needs not only the total reflectivity but also the distribution of reflected rays about the specular direction. Table 13.1.4 provides this information according to the formalism explained in Section 5.4.2. Rs(2ir) is the total or hemispherical reflectivity. The distribution of reflected rays is modeled either as a single Gaussian of width
376
Active Solar Collectors and Their Applications
TABLE 13.1.4 Properties of Reflector Materials3 Material Second surface glass Laminated glass (Carolina Mirror Co.) Laminated glass (Gardner Mirror Co.) Perpendicular to streaks Parallel to streaks Corning microsheet (vacuum chuck) Corning 03 17 glass (1.5
Measurement wavelength (nm)
Rt
500
0.92
600
0.92
500 800 500 550
0.92 0.88 0.92 0.77
K,(2ir)
7 for 9'a = 4 mrad
0.15
0.83
0.83
0.4
0.90
0.90
0.95
0.76
0.95
0.95
0.85
0.60
0.85
0.83
0.87
0.70
0.85
0.61
(mrad)
R2
(mrad)
0.4 <0.05 <0.05 1.1
0.18
6.2
mm) Metallized plastic films 3M Scotchcal 5400
3M FEK-163
Sheldahl aluminized teflon
Polished bulk aluminum Alcoa Alzak Perpendicular to rolling marks
500 600 700 900 500 600 700 900 400
0.86 0.86 0.82 0.84 0.86 0.86 0.82 0.84 0.73
0.90 0.78 0.86 0.86 1.4
0.15
12.1
500 700 900
0.80 0.80 0.81
1.3 1.6 1.4
0.07 0.04 0.03
30.9 39.8 31.4
670
0.66
0.39
0.21
9.7
505
0.56 0.45 0.70
0.42 0.53 0.24
0.33 0.42 0.17
10.1 9.8 7.7
407.5
0.62 0.58
0.29 0.46
0.27 0.29
7.1 9.0
498
0.65
0.37
0.23
16.1
498
0.67
0.43
0.21
18.5
550
0.44
1.4
0.43
10.3
407.5 Parallel to rolling marks Kingston Ind. Kinglux Perpendicular to rolling marks Parallel to rolling marks Metal Fabrications Bright aluminum
670 505
1.9
2.0 2.1 1.9
0.68
0.85
0.67 0.69
0.84
0.44
B
Rs(2ir) is hemispherical reflectivity,
vantage in applications with high concentration. For example, if the acceptance half-angle of the collector is 4 mrad, then only 0.67-0.69 of the light incident on Kinglux is reflected to the absorber even though the hemispherical reflectance is 0.85. As for costs, metalized plastic reflector films sell for about 8-35 $/m2 and
Practical Considerations
377
silvered glass for about 10-15 $/m2 (the cost of the silver itself makes only a small contribution because the layer can be very thin). Polished aluminum sheet like Alzak or Kinglux can be bought for about 25 $/m2. Of course the cost of a collector is not determined by the reflector surface alone. For example, an aluminum sheet has some rigidity by itself and may not need the extra expense of a shaped substrate. 13.1.4 Other materials Collector enclosures, collector support, insulation, seals, pipes, and heat transfer fluids are vital parts of a collection system. A serious failure of any of these parts can cause malfunction of the entire system. In the past when energy was cheap, little attention was paid to insulation. Following that tradition many of the early collectors were insulated with less than the economically optimal amount of insulation. Insulation is crucial if a collector is to have good thermal performance. In flat plate collectors, for example, both back and sides should have adequate insulation. Hot pipes and ducts and storage tanks should of course also be well insulated. Of course, insulating materials should have low thermal conductivities and low cost. They must also withstand the highest expected operating temperatures and thermal cycling. Insulation on outdoor pipes should be encased with metal cladding or UV inhibited PVC jackets; simply painting it will usually not provide sufficient protection [Meeker and Boyd, 1981]. Moisture can cause severe problems. For example, if fiberglass or open cell foam become wet, they can act like a heatpipe: water evaporates near the absorber and condenses near the enclosure, creating a thermal short circuit from absorber to ambient. Hence it is crucial to protect the insulation from external or internal leaks. Closed cell insulation is preferable because it avoids this problem. However, freon-blown insulation releases corrosive hydrofluoric or hydrochloric vapors if heated above 180°C. In fact many insulating materials outgas at elevated temperatures, and these gases may condense on the inside of the glazing thereby reducing the transmittance even if they are noncorrosive. Figure 13.1.3 displays the conductivity as a function of temperature for several insulating materials. Most conductivities increase rapidly with temperature. Therefore heat losses must be calculated with the conductivity at operating temperature. Glasswool (fiberglass) is the cheapest insulation and it can be used up to 370°C if it is made with a minimum of organic binders and preheated to drive off volatiles [Vresk, 1981]. Preformed seals (gaskets) and other sealing compounds are needed to form a seal between covers and collector or receiver enclosure. The sealing material should not outgas or lose its elasticity. For high-temperature applications (over 195°C) fluorocarbon seals are recommended, while for lower temperatures seals of silicone, acrylic, or acrylic copolymers are acceptable. Collector enclosures and supports must provide enough rigidity to keep the collector components in place. The rigidity requirement is particularly severe for collectors with high concentration. The enclosure should keep the
378
Active Solar Collectors and Their Applications
Figure 13.1.3 Conductivity as function of temperature for several insulating materials (From ASHRAE[ 1981], with permission).
inside dry and clean. It should allow rain and snow to run off. Common materials for collector enclosures or support are galvanized or painted steel, aluminum, fiberglass, plastics, wood and wood products, and concrete. Compatibility of materials is important. For instance, different metals in close contact with each other can cause severe electrolytic corrosion. Also the possibility of differential thermal expansion must be taken into account to prevent warping or breakage of glazing or absorber. In this context one might also emphasize the need for expansion tanks, and for expansion loops and/or bellows in the piping to accommodate thermal expansion. The materials and paints should be resistant to UV and other environmental influences. Wood is a risky choice as collector enclosure; it may outgas, warp, rot, or ignite. Last not least one needs to select the heat transfer fluid. Table 13.1.5 lists the properties of the most important candidates for solar application. Air costs nothing, is noncorrosive, and stable at all temperatures, but its heat transfer coefficient is low and the pumping power requirement high. Air is practical only for applications where hot air itself is the desired product. Water is the ideal heat transfer fluid as long as there is no risk of freezing or boiling. It offers the best possible heat transfer and pumping requirements
TABLE 13.1.5 Properties of Heat Transfer Fluids3 Commercial name (manufacturer) Propylene glycol Dowfrost (Dow)1 59 weight percent aqueous solution Sunsol 60 (Sunworks)2 60 weight percent aqueous solution Ethylene glycol Dowtherm SR-1 (Dow)3 59 weight aqueous solution Polyglycols UCON 500 (Union Carbide)
Chemical composition
Freezing point
Boiling point
Thermal conductivity (W/m °K)
Specific heat (kJ/kg °K)
Viscosity [10 m2/s]
6
Volume expansion (10~ 4 K~')
Specific gravity at °C
Propylene glycol K2HPO41%
-31"
102°
0.39
3.6 at 27°C
3.0 at 38°C 70 a t - 18°C
7.3
1.033 at 27°C
Propylene glycol inhibitor
-48°
109°
0.35
3.4 at 25°C
4.0 at 38°C 140 at- 20°C
7.5
1.055 at 25°C
Ethylene glycol K2HPO4 2%
-37°
110°
0.42
3.4 at 27°C
2.6 at 38"C 22 at - 18°C
6.3
1.074 at 20°C
Polyglycol 90% inhibitor
-37°
289° rec max temp
0.15
2.2 at 100°C
11.5atlOO°C 61 at38°C
7.9
0.98 at 100°C
TABLE 13.1.5 Properties of Heat Transfer Fluids3 (continued) Commercial name (manufacturer) Paraffinic hydrocarbon oils Caloria HT 43 (Exxon)4
Synthetic Hydrocarbon oils Brayco 888 HF (Bray Oil Co.)5 Silicone oils Syltherm 444 (Dow Corning)6 Water
Freezing point
Boiling point
Thermal conductivity (W/m °K)
Specific heat (kJ/kg °K)
Viscosity [10~6 m2/s]
Petroleum distillate moderate molecular weight, low aromatic content
-9.5°
311° rec max temp
0.13
2.1atlOO°C
Polymerized 1-decene
-85°
227° flash point
0.13
Polydimethysiloxane
-46°
> 315°
0°
100°
Chemical composition
CO
CO
Volume expansion (10~ 4 K-')
Specific gravity at °C
5.0atlOO°C 31 at38°C
10.0
0.80 at 100'C
2.3 at 25°C
4.5 at 100'C 22.5 at 38°C
4.8
0.83 at 15°C
0.14
1.6atlOO°C
7.0atlOO°C 20 at25°C
10.7
0.95 at 25°C
0.60
4.2 at 93°C
0.3 at 93°C
2.1
1.00 at4"C
Notes. Similar physical properties exist for ( I ) TJCAR 35, 50 percent solution (Union Carbide); Sunsafe 230, 50 percent solution (NPD Energy Systems); (2) Solar Winter Ban (CAMCO); Corona Solar Fluid (A. O. Smith); (3) UCAR 17, 50 percent solution (Union Carbide); (4) Mobiltherm 603 (Mobil); Thermia C (Shell); Dowtherm HP (Dow); (5) PAO-20E (Uniroyal): H-30 (Mark Enterprises); (6) SF-96 (General Electric). "From Sullivan [1980].
Practical Considerations
381
are low.3 Possible problems with corrosion can be prevented by adding inhibiting agents and following the recommendations in Table 13.1.6 concerning compatibility with absorber materials. The temperature limits of water can be extended by adding antifreeze. Typically one chooses a 50/50 mixture of water and glycol; this prevents freezing down to almost — 40°C and permits operation up to about 120°C. At higher temperatures glycols may decompose into corrosive byproducts. Hence it is advisable to check the condition of the antifreeze occasionally, in particular after an accidental overheating. Glycols are somewhat toxic, and U.S. building codes require a double wall heat exchanger between a drinking water supply and a collector with glycol. For operation in the range of 100-300°C one can use water only if one is willing to pay the price for high-pressure pipes. Otherwise one will resort to heat transfer oil. In their heat transfer characteristics oils are inferior to water, but certainly far better than air. Hydrocarbon oils are flammable and require precautions against fire. Also they tend to become very viscous or even freeze at ambient temperatures, thereby posing problems for system startup. Silicone oils avoid the flammability and viscosity problems of hydrocarbon oils, but they are more expensive, and they are even more leak prone. For applications above 300°C, the principal heat transfer fluids are air, steam, helium, molten salt, and liquid metal. Typical heat transfer coefficients for these fluids are listed in Table 9.1.1. For a more precise performance ranking, the reader is referred to Fried [1973], who has defined a convenient criterion, the heat transfer efficiency factor. 13.2 Installation 13.2.1
Collector location and orientation
The most obvious criterion for collector location is access to sunlight. Shading is acceptable only early in the morning or late in the afternoon when the collector would be able to deliver no or almost no useful energy anyway, but at other times shading should be minimized. If shading may be a problem, it is advisable to calculate sun angles and shadows cast by trees, buildings, etc. and carry out a system performance calculation for a clear day using the clear day insolation model of Section 3.5.2. As a general rule shading between 9 am and 3 pm should be avoided. In large collector arrays some shading from one collector row to the next is inescapable. Here a compromise must be made between losses due to shading on one hand and cost of land and longer piping runs on the other. The collector spacing can be expressed as ground cover ratio defined as ratio of total aperture and total 3 However, water has a high specific heat, and in some applications the heat capacity of a water-filled absorber can impose significant transient losses.
TABLE 13.1.6 Compatibility of Metals and Aqueous Solutions" Generally unacceptable use conditions Aluminum 1. When in direct contact with untreated tap 1. water with pH <5 or >9. 2. When in direct contact with liquid containing copper, iron, or halide ions. 2. 3. When specified data regarding the behavior of a particular alloy are not available, the velocity of aqueous liquids shall not exceed 4 ft/sec. Copper 1. When in direct contact with an aqueous 1. liquid having a velocity greater than 4 ft/ sec. 2. 2. When in contact with chemicals that can form copper complexes such as ammonium 3. compounds.
Generally acceptable use conditions When in direct contact with distilled or deionized water that contains appropriate corrosion inhibitors, When in direct contact with stable anhydrous organic liquids,
When in direct contact with untreated tap, distilled, or deionized water. When in direct contact with stable anhydrous organic liquids. When in direct contact with aqueous liquids that do not form complexes with copper.
Steel 1. When in direct contact with liquid having a 1. When in direct contact with untreated velocity greater than 6 ft/sec. tap, distilled, or deionized water. 2. When in direct contact with untreated tap, 2. When in direct contact with stable distilled, or deionized water with pH <5 or anhydrous organic liquids. > 12. 3. When in direct contact with aqueous liquids of 5 < pH < 12. Stainless steel 1. When the grade of stainless steel selected is 1. When the grade of stainless steel not corrosion resistant in the anticipated selected is resistant to pitting, crevice heat transfer liquid. corrosion, intergranular attack, and 2. When in direct contact with a liquid that is stress corrosion cracking in the in contact with corrosivefluxes. anticipated use conditions. 2. When in direct contact with stable anhydrous organic liquids. Galvanized steel 1. When in direct contact with water with pH 1. When in contact with water of pH > 7 <7or>12. but<12. 2. When in direct contact with an aqueous liquid with a temperature >55°C (131°F). Brass and other copper alloys Binary copper-zinc brass alloys (CDA 2XXX series) exhibit generally the same behavior as copper when exposed to the same conditions. However, the brass selected will resist dezincification in the operating conditions anticipated. At zinc contents of 15% and greater, these alloys become increasingly susceptible to stress corrosion. Selection of brass with a zinc content below 15% is advised. There are a variety of other copper alloys available, notably copper-nickel alloys, which have been developed to provide improved corrosion performance in aqueous environments. "From Kreiderand Kreith [1982]
382
Practical Considerations
383
ground area of a collector field. The optimal ground cover ratio depends on latitude and collector type; typical values for midlatitudes range from 0.20.4 for point-focus collectors and 0.4-0.6 for line-focus and nontracking collectors. The collector orientation should be chosen to maximize useful solar energy. Nontracking collectors should face due south, although some deviation from the rule is acceptable. At midlatitudes a collector azimuth (deviation from due south) up to 10° imposes little penalty, but azimuths in excess of 20° should be avoided. Close to the equator the tolerance for azimuth deviations is greater because the collector tilt is smaller. The optimal tilt for a fixed aperture is approximately equal to the latitude for year-round loads and approximately equal to latitude + 15° in the northern hemisphere for space heating. For parabolic troughs and linear Fresnel reflectors the basic goal is to keep the incidence angle small on the average. The optimal orientation for year-round loads is the polar mount (i.e., tracking axis northsouth with tilt equal latitude), but outside the equatorial regions, this may cause problems with collector supports, wind loading, and long piping connections. Hence one frequently prefers to keep the tracking axis horizontal. In this case the usual choices are to align the tracking axis either in the eastwest or the north-south direction. The north-south direction yields more yearly energy, but is highly nonuniform over the year (see Section 3.9); the optimal choice depends on the application. In any case the precise orientation of the tracking axis does not matter; just as with fixed collectors there is some tolerance. In fact a sloping ground facilitates draining of the collector. Also, a south-facing slope (in the northern latitudes) enhances energy collection. Linear Fresnel lenses are more demanding. If they are designed for a fixed tracking axis at all, then it must be the polar axis. If the concentration of a linear Fresnel lens is sufficiently high, then seasonal adjustments of the tracking axis are required or even full 2-axis tracking must be used to maintain proper focus. Frequently the choice arises between mounting the collectors on the ground or on the roof. This involves many factors. While unused roof area usually comes free, the roof may not be strong enough to bear the weight and wind loading of a collector, especially in a retrofit. In new buildings one can design the roof for the collector. When mounting collectors on the roof one should be careful to avoid leaks through roof penetrations needed for the collector supports. With ground mounting, on the other hand, the availability of land may be a problem (note that in most locations the cost of land is small compared to the cost of solar collectors). In residential applications, one usually places the collectors on the roof; for space heating a south-facing wall may also be excellent. There are additional considerations for the choice of the collector location. Accessibility is important for maintenance and repair. Collector, storage, and load should be near each other to minimize piping costs and pumping power. In particular, the hot lines should be kept short to reduce heat
384
Active Solar Collectors and Their Applications
losses.4 Environmental contaminants may be a problem in some industrial applications, and one should avoid placing a collector downwind from a source of dirty or corrosive effluents. On some roofs vapors from tar may condense on the collector and reduce its optical efficiency. Another selection criterion for the collector location may be the strength of wind. 13.2.2 Pumps and piping Careful attention should be given to the pipes/ducts of a solar energy system. The main issues are: flow diagrams, pipe diameter(s), pipe insulation; heat losses (both steady state and transient), and pumping power. Careful analysis and optimization of the piping subsystem can be a complicated task. The pumping power must be included with due attention for the higher value of mechanical energy as compared to thermal energy. Pumping power has often turned out to be unexpectedly large. With regard to flow diagram, there are several ways of interconnecting collector modules in an array: series, parallel, or some combination thereof. Series connection guarantees uniform flow but the pressure drop is likely to be prohibitive, especially in large arrays. For that reason one usually chooses parallel flow. Whenever collectors are put in series, the pressure drop should be checked. With parallel connections the main problem is nonuniformity of flow. This can be prevented by either making the feeder pipe large compared to the collector pipe(s) or else adding balancing valves to the individual collector modules. The latter approach is more labor intensive as the balancing may have to be checked periodically. To check whether the flow is indeed uniform in an actual installation, an infrared camera is most convenient; collectors with insufficient flow will be hotter than the others. With parallel connections there are two arrangements for the return pipe, direct return and reverse return, as shown in Fig. 13.2.1. Reverse return requires somewhat longer pipe connections but it automatically creates a nearly uniform flow distribution. In direct return networks the collectors on the far side of the inlet are likely to get starved, and balancing valves become necessary. Note that Fig. 13.2.1 also illustrates how inlet and outlet are located to minimize the length of the hotter pipe. Sufficient expansion loops or bellows should be provided to accommodate thermal expansion. Flex hoses should be installed in such a fashion that they suffer a minimum of torque and tension. Pipe supports and pump seals can cause unexpectedly large heat lakes. The piping connections at the storage tank should be properly placed to maximize storage utilization and efficiency. In particular, the outlet from tank to collector should be at the very bottom of the tank and the return from the collector should be at the top. Diffusers may be added to enhance stratification. In air systems with rock 4 In some cases heat losses from pipes and ducts may not be a real loss if they reduce the heat load of a building; but of course one has to make sure that this does not cause an increased cooling load.
Practical Considerations
385
Figure 13.2.1 Possible interconnection of collector modules in an array: (a) direct return; (b) reverse return (From Kutscher [1981]).
beds the inlet plenum from collector to storage should be large enough to ensure uniform flow; for this the plenum area should be at least 10% of the cross section of the bed. The rocks should be fairly smooth and uniform in size (% to 1 % times the nominal diameter). Pipe and tank materials, insulation, pumps, seals, and gaskets must be chosen to withstand the highest expected temperatures and pressures. (Note that water mains pressure can be several atmospheres.) To increase reliability one can install several pumps. In the right size combination they can provide a good low-cost approximation to variable speed pumps. Leaks can be a problem anywhere in a solar thermal system. Particularly prone to leakage are systems with air and systems with oil. The reason for air systems has to do with the construction of air ducts because of the need for large cross sections. They are made of sheet metal, and the seams are rarely airtight. Heat transfer oils have a low surface tension and they are notorious for their ability to seep through the finest cracks. Needless to say they are also a fire hazard. Several kinds of valves may be necessary. There must be a sufficient number of pressure relief valves to prevent undue pressure buildup in the collector field. They must be positioned so they do not discharge fluid onto collectors or people. Shut-off valves are needed to facilitate repairs. In some systems the collectors are higher than the storage, and at night, warm fluid
386
Active Solar Collectors and Their Applications
from storage could thermosyphon to the collectors; this can be prevented by check valves. Finally before starting a system one should flush out any dirt that might have gotten into pipes, collectors, tank, or filter. 13.2.3 Safety and protection of equipment Freeze protection of water systems has already been discussed in Section 12.2.1. Water is the only practical heat transfer fluid that expands upon freezing. But even without such expansion the solidifying of the heat transfer fluid can cause problems. Certain high-temperature systems use molten salt or oil as heat transfer fluid, and at night the fluid in the pipes may solidify or become very viscous. This may paralyze the pump and allow overheating in the receiver, or the strain on the pump may cause excessive wear. To forestall this possibility one either adds trace heaters to the pipes or else one keeps circulating the fluid at all times (provided the heat losses from the receiver are small enough); or else one selects a different heat transfer fluid. One also needs to worry about overheating of the collectors. This can happen under several circumstances: during construction and before startup, during pump or power failures, and in space heating systems during the summer season. Overheating must be prevented unless the collector and the fluid are certified to be stagnation proof. In any case extended periods of stagnation during construction should be avoided by proper scheduling; or better, by keeping the aperture covered or in the shade. In space heating systems the collector could be vented to ambient in summer (if it is an air system), or else to an extra heat exchanger; otherwise a collector cover may be necessary. Overheating during brief pump failures may not be a serious problem in flat plate collectors, although antifreeze in the collector may degrade. Also, after boiling has occurred and the pressure relief valve has opened, harmful underpressure may develop when the collector cools down again. A particularly destructive phenomenon can occur with overheated evacuated collectors; if cold liquid reaches the tubes before they have cooled down, the thermal shock will break the tubes. Without thermal shock most of the evacuated collectors on the market today can survive stagnation for extended periods, even though their stagnation temperature may exceed 300°C. With tracking collectors the protection against stagnation is easy. One simply turns the reflector away from the sun. Of course this will not work if tracker and pump fail at the same time, as they will during a power outage. If the heat capacity of the receiver is large enough the sun will be out of focus before any harm is done. This is likely to be the case, with the exception of collectors with east-west tracking axis for which, at equinox, the sun remains in focus all day. If overheating of a tracking collector is possible, then an emergency mechanism is advisable to turn the collector out of the sun. Tracking collectors should also be turned to a safe stowing position during
Practical Considerations
387
dangerously high winds (in excess of 30-50 m/sec depending on collector construction). A safety concern with concentrating collectors arises from high flux levels near the receiver while the collector is being turned into or out of the sun. Insulation, pipes, and other parts in the vicinity of the absorber could get burnt unless they are adequately shielded. People who are near concentrators should wear protective glasses and be extremely careful. Central receivers should be located away from flight paths of airplanes or balloons. In some areas fence may be advisable to keep inadvertent intruders or worse, vandals, from getting on-site. High-temperature systems with oil present a fire hazard and proper precautions should be taken. For example, the operating personnel should be properly trained, and a central switch, accessible to firemen, should allow turning off the flow and stowing the reflectors at the same time.
13.3 MAINTENANCE AND CLEANING 13.3.1 Maintenance Solar energy systems perform best if they are properly maintained and cleaned. The amount of maintenance depends on the complexity of the system and on collector type. In general, tracking collectors and high-temperature systems require more attention. For focusing collectors the most important maintenance problem is cleaning. Cleaning is discussed in the following section. As always there is a trade-off between the cost of maintenance and the benefit of improved system performance and system life. Any system should be inspected after accidental stagnation. Antifreeze and heat transfer oils decompose at high temperatures and may need to be replaced if stagnation has occurred. Regular inspections are advisable to check temperatures, pressures, valves, and controls, to test level and condition of heat transfer liquid, to look for damage or leaks, and to do any adjustments, cleaning, or repair that may have become necessary. Inspections are all the more necessary as long as the technology is new and long term operating experience is scant. Even for relatively simple residential systems yearly inspections are recommended by Solar Age [1982]. In systems with heat transfer oils one needs to guard against leaks because of the fire hazard. Needless to say, if a system is shut down for maintenance it cannot collect energy; hence it is preferable to schedule repair and maintenance for overcast days or periods without load or at night. For industrial installations, the size of a solar system has an interesting implication for the attitude of the maintenance personnel: if the system is very small compared to the load, it may not be taken seriously and receive the proper attention. Proper training of the maintenance personnel is important.
388
13.3.2
Active Solar Collectors and Their Applications
Cleaning
The need for collector cleaning varies widely with collector type and location. To understand the effect of dirt on different collector types, let us take a closer look at the interaction of light with dust particles. Dust causes both absorption and scattering. Most dust particles are very small and they affect the passage of light as much by scattering as by absorption [Berg, 1978]. Since the scattered radiation is distributed over a wide range of angles it will miss the receiver of a focusing collector. In a flat plate collector, by contrast, even the radiation scattered by dust on the cover (at least the radiation scattered into the forward hemisphere) will reach the absorber. For that reason flat plate collectors are quite insensitive to dust; they may function surprisingly well even when they look terrible. Besides, most flat plate collectors are deployed at a tilt where rain provides adequate cleaning. In residential applications people rarely bother to clean flat plate collectors. Industrial systems are larger and economies of scale may make cleaning of flat plate collectors cost effective, at least in dusty environments. Concentrating collectors with large acceptance angle (e.g., CPCs) lose some of the scattered radiation and are intermediate between flat plate and focusing collectors in their sensitivity to dust. The effect of dust depends on optical design in yet another way. On an exposed second surface reflector a light ray has to penetrate the dust layer twice on its way to the receiver. The thickness of the protective coating in front of the reflective metal layer itself is large compared to the size of dust particles. Therefore the scattering and absorption on the way to the metal layer is uncorrelated with the scattering and absorption on the way out from the metal layer. Hence the loss of radiation with an exposed second surface reflector is twice as large as it would be if light had to pass the dust layer only once. The latter is the case with Fresnel lenses (assuming the back of the lens stays clean) and with parabolic dish collectors that are completely enclosed inside inflated transparent bubbles. Several factors determine the accumulation of dust on collectors. The deposition of dust depends on wind velocity, surface shape, and particle size. The bonds that dust or dirt particles form with a surface may be chemical or physical. There is a tendency for the bond to become stronger with time. Atmospheric moisture plays an important role in this bonding process because moisture brings a dust particle closer to the surface, thereby increasing the forces of adhesion; the increased bond persists even after the moisture has evaporated again [Berg, 1978]. Dust accumulation is most severe at night when dew condenses on airborne dust, and makes it settle on a collector, with the water acting like a glue. A light rain in the desert causes similar problems if the precipitation is only enough to collect dust from the air but not enough to rinse off the collector. For this reason it is important to stow reflectors so they do not face upwards at night or during dust storms. For heliostats the vertical position, facing out of the wind, is also acceptable. Fixed reflector systems (i.e., the hemispherical bowl and the cylindrical slats [See Section 7.4]) suffer in this regard and require extra cleaning.
Practical Considerations
389
For a more quantitative discussion of the effect of dirt on reflectors we cite a study by Freese [1978]. The specular reflectances of mirrors were monitored through nine months of outdoor exposure and different cleaning frequencies. Differences in reflector elevation and mounting angle (other than inverted) were found to have no significant effect on the rate of dirt and dust accumulation. Figures 13.3.1a-13.3.1c show reflectance as a function of time for mirrors cleaned every 2, 6, and 12 days. The sawtooth pattern indicates that reflectance deteriorates rapidly after cleaning but is restored to a fairly constant maximum with each cleaning. Figure 13.3. Id depicts the same relationship for a mirror never cleaned except by natural weather conditions. It was found that specular reflectance decreased most rapidly on just-cleaned mirrors, that light rain with windy, dusty conditions could decrease reflectance up to 15-20 percentage points, and that rain on a newly cleaned mirror slightly decreased its reflectance, but rain and melting snow on dirty mirror surfaces cleaned them well. In fact, snow conditions could increase reflectance to within 0.01 reflectance units of the value obtained after ultrasonically cleaning the mirrors in a laboratory. Ultrasonic cleaning, of course, is not a practical solution to problems of dirt and dust accumulation on reflectors. Realistic cleaning methods include high-pressure water sprays, sprays of commercial detergents, and wiping with soap and water, all followed by a water rinse. Wiping may not be feasible if access to reflectors is blocked by supports, receivers, or guy wires. Cleaning large reflector arrays with detergent may not be environmentally sound and it does not appear to be significantly more effective than cleaning with plain water [Pettit and Butler, 1977]. High-pressure plain water sprays of 3.5 MPa are recommended; they can recover about 95% of the reflectance lost due to particle accumulation [Berg, 1978]. Either an automatic sprinkler system or a spray truck with water tanks can supply the water jets. To minimize particle accumulation between cleanings, reflectors should be stowed facing vertically or downward when possible, and they can be rinsed after cleanings with an antistatic solution [Champion, 1978]. Dust and dirt on an exposed surface is relatively easy to clean off. Dirt on the inside of a glazing or on the absorber poses a more serious problem. Dirt on the absorber affects both absorptivity and emissivity. The absorptivity of dirt tends to be lower than that of most absorbers. Furthermore the emissivity of most dirt is high, and it can significantly affect the performance of a selective absorber. To illustrate this point, suppose that one-tenth of a selective absorber with eabs = 0.1 is covered with dust, grease, or fingerprints of emissivity edirt = 0.9. Then the average emissivity is almost doubled:
The absorber and inside of the glazing are difficult or impossible to clean. Therefore these surfaces should be adequately protected. In a nonevacuated collector hermetic sealing seems impractical because of the thermal expan-
Figure 13.3.1 Specular reflectance versus time for a mirror under several different cleaning cycles: (a) cleaning every 2 days; (b) cleaning every 6 days; (c) cleaning every 12 days; (d) no cleaning (From Freese [1978]). 390
Figure 13.3.1
(continued)
391
392
Active Solar Collectors and Their Applications
sion of the air inside the collector; instead, vent holes should be covered with dust niters. 13.3.3
Tracking
The performance of tracking collectors depends on the tracking accuracy that can be maintained in actual operation. Many of the tracking collectors installed during the 1970s fared rather poorly in this regard [Gee, 1982]. A frequent problem was caused by the inability of the first generation of tracking sensors to function well under variable insolation. They could be adjusted to work correctly for a fixed insolation level, but with changing beam or diffuse insolation they would become erratic. Sometimes these trackers would chase the silver lining of a cloud instead of the sun. Another common problem with the early trackers had to do with misalignment between sensor and collector. Some systems tried to bypass the problems caused by a sun sensor and instead used a computer to calculate the position of the sun at any moment. However, this latter approach requires very careful determination of the orientation of the collector; otherwise there will be a serious systematic tracking error. Apparently these early problems have been more or less solved. When measuring the accuracy of recent commercial trackers for parabolic troughs, Gee [1982] found that correct tracking could be maintained within about 1 mrad. With heliostats, tracking accuracies in the range of 1-2 mrad have been demonstrated at the Central Receiver Test Facility in Albuquerque, NM [Holmes, 1982]. Tracker improvements have been achieved by more sophisticated electronics and by changes in sensor design, thereby reducing the impact of diffuse insolation. Also some new trackers have been developed that measure directly the flux at the receiver, thus eliminating many of the alignment problems experienced with trackers mounted on the aperture. While tracking accuracy of the new trackers is best for high levels of beam irradiance, it remains good down to beam irradiance levels of 200 W/
Figure 13.3.2 Effect of tracking errors on annual collectible energy (From Gee [1983]).
Practical Considerations
393
m2, and only the beam-normal irradiance seems to matter, not the angle of incidence or the diffuse irradiance [Gee, 1982]. To provide a measure of the importance of tracking accuracy, Fig. 13.3.2 shows the decrease in annual collected energy as a function of tracking error for parabolic troughs. Curves are shown for several values of concentration ratio C and mirror contour error amMam. The higher the concentration, the greater the sensitivity to tracking errors. Tracking errors of 5 mrad are quite serious; they cause a loss of about 1% for C = 15 and 7% for C = 35. But if the tracking error is less than 1 mrad, then the loss is negligible (a fraction of 1%) even for C as high as 35. REFERENCES Atkinson, B and Caesar, R. 1983. "The Volkspanel Model T." Solar Age (April):33. ASHRAE. 1981. Handbook of Fundamentals. Atlanta, GA: Society of Heating, Refrigeration and Air Conditioning Engineers. Beauchamp, E. K. 1975. "Low Reflectance Films for Solar Collector Cover Plates." Report SAND 75-0035. Albuquerque, NM: Sandia Laboratories. Berg, R. S. 1978. "Heliostat Dust Buildup and Cleaning Studies." Report SAND 780510. Albuquerque, NM: Sandia Laboratories. Butler, B. L. and Claassen, R. S. 1980. "Survey of Solar Materials." Trans. ASME/ J. Solar Energy Eng. 102:175. Call, P. and Masterson, K. 1978. "Absorber Surfaces and Reflective Materials." Proc. of the Solar Thermal Concentrating Collector Technology Symposium, Report SERI/TP-34-048. Golden, CO: Solar Energy Research Institute. Call, P. J. 1979. "National Program Plan for Absorber Surfaces R & D." Report SERI/TR-31-103. Golden, CO: Solar Energy Research Institute. Cash, M. 1978. "Learning from Experience." Solar Age 3:11. Champion, R. L. 1978. "Cleaning and Maintenance." Proc. of the Solar Thermal Concentrating Collector Technology Symposium. Report SERI/TP-34-048. Golden, CO: Solar Energy Research Institute. Clements, L. D. 1978. "High Temperature Receiver Materials Performance." Proc. of the Solar Thermal Concentrating Collector Technology Symposium. Report SERI/TP-34-048. Golden, CO: Solar Energy Research Institute. Freese, J. M. 1978. "Effects of Outdoor Exposure on the Solar Reflectance Properties of Silvered Glass Mirrors." Report SAND 78-1649. Albuquerque, NM: Sandia Laboratories. Fried, J. R. 1973. "Heat Transfer Agents for High Temperature Systems." Chem. Eng. May 28. Gee, R. C. 1982. "An Experimental Performance Evaluation of Line Focus Sun Trackers." Report SERI/TR-632-646. Golden, CO: Solar Energy Research Institute. Gerwin, H. 1978. "Line Focus Receiver Technology." Proc. of the Solar Thermal Concentrating Collector Technology Symposium. Report SERI/TP-34-048. Golden, CO: Solar Energy Research Institute. Godolphin, D. 1982. "Reliability and Durability in Solar Energy Systems." Solar Age 7:10. Harding, G. L. and Window, B. 1980. "Graded Metal Carbide Selective Absorbing
394
Active Solar Collectors and Their Applications
Surfaces Coated onto Glass Tubes by a Magnetron Sputtering System." J. Vac. Sci. Technol. 16:2101. Harding, G. L. and Moon, T. T. 1981. "Calorimetric Measurement of Absorptance and Emittance of the Sydney University Evacuated Collector." Solar Energy 26:281. Holmes, J. T. 1982. "Heliostat Operation at the Central Receiver Test Facility." ASMEJ. Solar Energy Eng. 104:133. Kutscher, C. F. 1981. "Design Considerations for Solar Industrial Process Heat Systems." Report SERI/TR-632-783. Golden, CO: Solar Energy Research Institute. Kutscher, C. F. et al. 1982. "Design Approaches for Solar Industrial Process Heat Systems." Report SERI/TR-253-1356. Golden, CO: Solar Energy Research Institute. Lampert, C. M. 1982. "Durable Innovative Solar Optical Materials—The International Challenge." In Optical Coatings for Energy Efficiency and Solar Applications, " p. 324. Soc. of Photo-Optical Instrumentation Engineers. Lampert, C. M. and Washburn, J. 1979. "Microstructure and Optical Properties of Black Chrome Before and After Exposure to High Temperatures." Proc. of the Second Annual Conference on Absorber Surfaces for Solar Receivers. Report SERI/TP49-182. Golden, CO: Solar Energy Research Institute. Leonard, J. A. 1978. "Linear Concentrating Solar Collectors—Current Technology and Applications." Proc. of the Solar Thermal Concentrating Collector Technology Symposium. Report SERI/TP-34-048. Golden, CO: Solar Energy Research Institute. Lind, M. A., Pettit, R. B., and Masterson, K. D. 1980. "The Sensitivity of Solar Transmittance, Reflectance and Absorptance to Selected Averaging Procedures and Solar Irradiance Distributions." ASME J. Solar Energy Eng. 102:34. Jorgensen, G. 1979. "Long-Term Glazing Performance." Report SERI/TP-31-193. Golden, CO: Solar Energy Research Institute. Kreider, J. and Kreith, F. 1982. Solar Heating and Cooling: Active and Passive Design, 2nd ed. New York: McGraw-Hill. Meeker, J. and Boyd, L. 1981. "Domestic Hot Water Installations: The Great, the Good, and the Unacceptable." Solar Age 6:10. Modern Plastics Encyclopedia. Published annually, New York: McGraw-Hill. Muenker, A. H. 1979. "High Temperature Solar Absorber Paints." Proc. of the Second Annual Conference on Absorber Surfaces for Solar Receivers. Report SERI/TP49-182. Golden, CO: Solar Energy Research Institute. Peterson, R. E. and Ramsey, J. W. 1975. "Thin Film Coatings in Solar-Thermal Power Systems."/. Vacuum Sci. Technol. 12:174. Pettit, R. B. 1977. "Characterization of the Reflected Beam Profile of Solar Mirror Materials." Solar Energy 19:733. Pettit, R. B. and Butler, B. L. 1977. "Semiannual Review ERDA Thermal Power Systems, Dispersed Power Systems, Distributed Collectors, and Research and Development: Mirror Materials and Selective Coatings." Report SAND 77-0111. Albuquerque, NM: Sandia Laboratories. Rabl, A. 1977. "Prisms with Total Internal Reflection as Solar Reflectors." Solar Energy 19:555. Seraphin, B. O. and Meinel, A. B. 1976. "Photothermal Solar Conversion and the Optical Properties of Solids." In Optical Properties of Solids: New Developments, Seraphin, B. O., editor. Amsterdam: North Holland. Seraphin, B. O. and Carver, G. E. 1979. "Chemical Vapor Deposition of Refractory
Practical Considerations
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Metal Reflectors for Spectrally Selective Solar Absorbers." Annual Report: May 1, 1978 to April 30, 1979. Tucson, AZ: Optical Sciences Center, University of Arizona. Smestad, G. and Hamill, P. 1982. "Concentration of Solar Radiation by White Painted Transparent Plates." Appl. Opt. 21:1298. Solar Age. 1982. "Questions and Answers." (August): 8. Solar Energy Corporation. Specifications Sheet for Solkote Hi/Sorb-I Selective Solar Coating. Solar Energy Corporation, Box 3065, Princeton, NJ 08540. Sparrow, E. M. and Cess, R. D. 1978. Radiation Heat Transfer. New York: McGrawHill. St. Amand, R. P., Williams, J. C, and Farrauto, R. J. 1979. "Improved Solar Absorber Coatings." Proc. of the Second Annual Conference on Absorber Surfaces for Solar Receivers. Report SERI/TP-49-183. Golden, CO: Solar Energy Research Institute. Sullivan, T. 1980. "Solar Heat Transfer Fluids." Solar Age (December):!!. Tortorello, A. J. and Wolf, R. E. 1979. "Outgassing Studies of Some Solar Absorber Coatings." Proc. of the Second Annual Conference on Absorber Surfaces for Solar Receivers. Report SERI/TP-49-182. Golden, CO: Solar Energy Research Institute. USDOE. 1978. "National Solar Heating and Cooling Demonstration Program: Project Experience Handbook." Report DOE/CS-0045/D. U.S. Department of Energy. Vitko, J., Jr. 1978. "Optical Studies of Second Surface Mirrors Proposed for Use in Solar Heliostats." Report SAND 78-8228. Livermore, CA: Sandia Laboratories. Vresk, J. et al. 1981. "Final Reliability and Materials Design Guidelines for Solar Domestic Hot Water Systems." Report ANL/SDP-11. Argonne National Laboratory. Winston, R. 1976. "Dielectric Compound Parabolic Concentrators." Appl. Opt. 15:291.
14. ECONOMIC ANALYSIS
Money does not talk, it swears. Bob Dylan
A solar energy system can be considered an investment that produces revenue in the form of fuel savings. The purpose of the economic analysis is to evaluate the profitability of a solar energy project and to compare it with alternative investments. The alternatives might include an oil-fired boiler, a heat recovery unit, a cogeneration system, or nonenergy investments such as stocks, bonds, and real estate. The economic analysis' provides the criterion not only for deciding whether to build a solar energy system but also for optimizing its design. The simplest way to deal with inflation is based on constant currency and real rates of return. This is explained in Section 14.1. The basic tools for the economic analysis are developed in Section 14.2. They are applied in Section 14.3 to derive the principal criteria for evaluating the economic attractiveness of an investment. Section 14.4 discusses life cycle costs and includes operation and maintenance expenses. Section 14.5 addresses complications that arise from debt financing and tax laws and for which an analysis in current dollars is more appropriate. 14.1 CONSTANT CURRENCY When analyzing future cash flows one cares about their real value, not about their nominal value in a currency that has been eroded by inflation. If one describes future cash flows in terms of inflating dollars (also called current dollars), one must therefore correct them for inflation. Inflation affects costs and benefits alike, leaving the balance unchanged. In this sense the inflation rate is a spurious variable. The simplest way to deal with inflation is to eliminate it from the analysis by working with constant currency and real discount rates. Real rates are defined as the difference between market rates and the inflation rate. For example, suppose in 1980 a sum of $1000 is 'The analysis might be based on strictly financial considerations or it might attempt to include other considerations such as reduced pollution, reduced dependence on fossil fuel supplies, or, for the nation as a whole, reduced risks associated with oil imports or with CO2 buildup in the atmosphere.
396
Economic Analysis
397
invested in a certificate of deposit bearing interest at rmarket = 1 5% whil inflation rate is / = 1 2%. After 1 year the investment has grown to $ 1 1 50 i 1981 dollars, but the value in 1980 dollars is $1150/1.12 = $1027. The value has grown at a real interest rate rrea, = 2.7%. The real rate is given by
or
As proved in Section 14.5.4, for an equity investment without taxation an analysis in constant currency is exactly equivalent to one in inflating currency.2 The former has the advantage of being simpler and more transparent because one needs to specify one variable less. Above all, the inflation rate is difficult to predict, while real rates are fairly well known. For example, from 1955-1980 the real interest rate on first rate corporate bonds has consistently been close to 2.2%, despite large fluctuations of the inflation rate [Jones, 1982]. The high real interest rates of the early 1980s are probably an anomaly. Riskier investments (e.g., the stock market) offer higher returns, but they, too, tend to be fairly constant in constant dollars. On the other hand, when an investment is financed by a loan or when the investor can claim tax benefits from depreciation there may be numerical differences between a constant currency analysis and one in inflating currency. Also, loan payments are usually arranged to have fixed amounts in inflating currency (at least that is current practice in the U.S.). The real value of the individual annual loan payments is different between two methods of analysis.3 For these reasons we use current dollars in the formulas for debt financing and depreciation in Section 14.5. 14.2
14.2.1
COMPARING PRESENT AND FUTURE COSTS
The time value of money
A dollar that is to be paid in the future does not have the same value as a dollar available today. This is true even if there is no inflation or if one is talking about constant currency. Since a dollar that is available today can 2 For that reason the equations of Section 14.3 can be used with real or with market rates, provided one is consistent and uses only one kind throughout. 3 If transactions have a fixed amount in inflating currency then their real value decreases with time. In the limit of very high inflation a long term loan becomes meaningless if its repayment schedule is fixed in inflating currency because then most of its value would be repaid during the first year. The fixed charge rate J^r of Eq. (14.4.9) is different in constant and in inflating currency.
398
Active Solar Collectors and Their Applications
be invested and bear interest, its future value is increased by the interest. Suppose an amount P (present worth) is invested at an interest rate r with annual compounding of interest. After the first year the value has increased to P(\ + r), after the second year to P(l + r)2, etc. Hence the future worth F after N years is
The ratio P/F is called the present worth factor, and its inverse is the compound amount factor. In accordance with the recommendations of the Engineering Economy Division of the American Society for Engineering Education we shall use the mnemonic notation
Analogous expressions can be written for other factors that are useful for economic analysis. In particular the compound amount factor is
Numerical values are listed in Table 14.2.1. These formulas are the basic tools for comparing payments that are made at different times. 14.2.2 How to choose the rate for discounting the future The present worth factor reduces or discounts the value of future transactions; hence, the rate r can be thought of as a discount rate. The higher r, the lower the present value of a future amount of money. This discounting of the future is a consequence of the productivity of capital. If all capital investment yielded the same rate of return or interest rate r, then all future cash flows would have to be discounted at the rate r. But while the need for discounting is clear, the choice of the rate is not, because different investments yield different returns. A few examples will illustrate how the discount rate may be chosen in a particular situation. First consider a homeowner who has enough cash in his savings account to purchase a solar water heater. Instead of buying the water heater he could leave the money in his savings account. If he does buy the water heater he foregoes the interest he could otherwise earn from his savings account. Hence his discount rate is the interest rate of his savings account. If he had no savings, he would have to borrow money for the water heater. In that case his discount rate would be the interest rate of his loan. Loan interest rates are generally higher than the rales for savings accounts. The situation is more complex if there are several investment alternatives. For example, a homeowner might invest his money in real estate,
399
Economic Analysis TABLE 14.2.1 Present Worth Factor (P/F, r, N) = (1 + r)-"toT Discount Rate r and Number of Years N
(%) 1
2
3
4 5 6 7 8 9 10 11
12 13 14
15 16 17 18 19 20
N
5
10
15
20
25
30
0.9515 0.9057 0.8626 0.8219 0.7835 0.7473 0.7130 0.6806 0.6499 0.6209 0.5935 0.5674 0.5428 0.5194 0.4972 0.4761 0.4561 0.4371 0.4190 0.4019
0.9053 0.8203 0.7441 0.6756 0.6139 0.5584 0.5083 0.4632 0.4224 0.3855 0.3522 0.3220 0.2946 0.2697 0.2472 0.2267 0.2080 0.1911 0.1756 0.1615
0.8613 0.7430 0.6419 0.5553 0.4810 0.4173 0.3624 0.3152 0.2745 0.2394 0.2090 0.1827 0.1599 0.1401 0.1229 0.1079 0.0949 0.0835 0.0736 0.0649
0.8195 0.6730 0.5537 0.4564 0.3769 0.3118 0.2584 0.2145 0.1784 0.1486 0.1240 0.1037 0.0868 0.0728 0.0611 0.0514 0.0433 0.0365 0.0308 0.0261
0.7798 0.6095 0.4776 0.3751 0.2953 0.2330 0.1842 0.1460 0.1160 0.0923 0.0736 0.0588 0.0471 0.0378 0.0304 0.0245 0.0197 0.0160 0.0129 0.0105
0.7419 0.5521 0.4120 0.3083 0.2314 0.1741 0.1314 0.0994 0.0754 0.0573 0.0437 0.0334 0.0256 0.0196 0.0151 0.0116 0.0090 0.0070 0.0054 0.0042
stocks, or bonds, each with a different rate of return and a different risk. Industrial and commercial investors usually have quite a range of investment opportunities. In such cases the appropriate discount rate for the analysis of a solar investment is the rate of return for investments of comparable risk that would be made if the money were not spent on the solar energy system. When a project is examined from the point of view of society, the discount rate should be based on the rate of return for society as a whole. This rate is called the social discount rate. It is the subject of much controversy, touching as it does on delicate problems such as intergenerational equity. A high discount rate de-emphasizes the importance of future costs or benefits; in other words it favors the present generation over future generations. The above discussion shows that the discount rate is easy to determine when there is a single alternative investment (or, equivalently, a single source of funds) such as a savings account. But when there are several investments with different rates of return, the choice of the discount rate may involve a difficult evaluation of risk. Hence the question arises whether one could base the discount rate on the productivity (i.e., rate of return) of the solar investment itself rather than on the productivity (i.e., interest rate) of money in general. This approach is indeed taken for many industrial enterprises. One writes down the formula for the economic analysis in terms
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Active Solar Collectors and Their Applications
of an unspecified discount rate r and determines r at the end by setting the cost of solar energy (which depends on r) equal to the cost of conventional energy. This method, called internal rate of return method, is explained more fully in Section 14.3.2. 14.2.3 Levelizing It is convenient to express costs or revenues that occur once or in irregular intervals as equivalent equal payments in regular intervals. This is called levelizing. It is a useful technique because regularity facilitates understanding and planning. To explain this technique let us consider an example. A bank wants loan payments to be arranged as a series of equal monthly or yearly installments. Suppose a loan of value P is repaid in equal annual payments A over N years. (The fraction of A that pays for interest varies with time, as discussed in Section 14.4.1, but for the purpose of levelizing we need not worry about that.) To derive the formula for A, consider first a loan of value Pn that is repaid with a single payment Fn at the end of n years. With n years worth of interest on the amount Pn the payment is
In other words, the loan amount Pn equals the present value of the future payment F,,:
A loan that is repaid in N equal installments can be considered as the sum of TV loans, one for each year, the nth loan being repaid in a single installment A at the end of the «th year. Hence the value P of the loan equals the sum of the present values of all loan payments:
Using the well known formula for geometric series one can evaluate this sum, with the result
Even though derived for the specific case of a bank loan, the result is perfectly general and relates any single present value P to a series of equal payments A, given the discount rate r and the number of payments N. When r vanishes, Eq. (14.2.5) is singular but its limit as r -* 0 is easily found to be 1//V, reflecting the fact that the present values of the loan payments become equal. Following the notation introduced above we designate the ratio of A
Economic Analysis
401
and P by
It is called capital recovery factor. Numerical values4 are listed in Table 14.2.1. The inverse
is known as series present worth factor since P is the present value of a series of equal payments A. EXAMPLE 14.2.1
What are the annual payments for a loan P = $1000 at interest rate r = 8% over N = 20 years? SOLUTION
The capital recovery factor is (A/P,r,N) = 0.1019, from Table 14.2.2. Hence the annual payments are $101.90, approximately one-tenth of the loan amount. The following example shows how present worth factors and capital recovery factors can be used. Suppose a solar energy system with life N needs a major overhaul in the nth year at a cost C. What is the levelized cost of this overhaul? The present value of the cost is
Hence the levelized cost is
From Eqs. (14.2.2) and (14.2.6) one finds
4
For the limit of long life, N -> oo, it is worth noting that (A/P, r, N) -» r if r > 0.
402
Active Solar Collectors and Their Applications TABLE 14.2.2 Capital Recovery Factor (A/P, r, N) = r/[\ - (1 + r)~N] for Discount Rate r and Life N of Investment
N
(%) -4
3
-2 J
0 1 2 3 4 5 6 7
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
5
10
0.1767 0.1824 0.1882 0.1940 0.2000 0.2060 0.2122 0.2184 0.2246 0.2310 0.2374 0.2439 0.2505 0.2571 0.2638 0.2706 0.2774 0.2843 0.2913 0.2983 0.3054 0.3126 0.3198 0.3271 0.3344 0.3418 0.3492 0.3567 0.3642 0.3718
0.0793 0.0843 0.0893 0.0946 0.1000 0.1056 0.1113 0.1172 0.1233 0.1295 0.1359 0.1424 0.1490 0.1558 0.1627 0.1698 0.1770 0.1843 0.1917 0.1993 0.2069 0.2147 0.2225 0.2305 0.2385 0.2467 0.2549 0.2632 0.2716 0.2801
15
20
25
30
0.0474 0.0518 0.0565 0.0615 0.0667 0.0721 0.0778 0.0838 0.0899 0.0963 0.1030 0.1098 0.1168 0.1241 0.1315 0.1391 0.1468 0.1547 0.1628 0.1710 0.1794 0.1878 0.1964 0.2051 0.2139 0.2228 0.2317 0.2408 0.2499 0.2591
0.0317 0.0358 0.0402 0.0449 0.0500 0.0554 0.0612 0.0672 0.0736 0.0802 0.0872 0.0944 0.1019 0.1095 0.1175 0.1256 0.1339 0.1424 0.1510 0.1598 0.1687 0.1777 0.1868 0.1960 0.2054 0.2147 0.2242 0.2337 0.2433 0.2529
0.0225 0.0263 0.0304 0.0350 0.0400 0.0454 0.0512 0.0574 0.0640 0.0710 0.0782 0.0858 0.0937 0.1018 0.1102 0.1187 0.1275 0.1364 0.1455 0. 1 547 0.1640 0.1734 0.1829 0.1925 0.2021 0.2118 0.2215 0.2313 0.2411 0.2509
0.0166 0.0201 0.0240 0.0284 0.0333 0.0387 0.0447 0.0510 0.0578 0.0651 0.0726 0.0806 0.0888 0.0973 0.1061 0.1150 0.1241 0.1334 0.1428 0.1523 0.1619 0.1715 0.1813 0.1910 0.2008 0.2107 0.2206 0.2305 0.2404 0.2503
Thus any future transaction can be expressed as an equivalent series of equal annual payments. EXAMPLE 14.2.2
A system has salvage value S = $1000 at the end of its useful life of N = 20 years. What is the equivalent levelized annual salvage value if the discount rate is r = 8%? SOLUTION
From Eq. (14.2.8) with n = TV and C = S we get, using Tables 14.2.1 and 14.2.2,
A = 0.1019 X 0.2145 S = 0.02186 S = $21.86.
Economic Analysis
403
Some payments can be assumed to increase or decrease at a constant annual rate. Fuel prices, in particular, are likely to exhibit real growth; i.e., to increase faster than general inflation in the foreseeable future. It is convenient to replace a growing or decreasing cost by an equivalent constant or levelized cost. Suppose, for example, that the price of energy is pe at the start of the first year, escalating at a real rate re. Including escalation and discount factors, one obtains the present value Pe of the total energy payments for a constant annual energy consumption Q:
(This assumes that payments are made at the end of each year.) Introducing a new variable fe
or
one can write the present value as
The levelized annual payment corresponding to this present value is obtained by multiplying by (A/P, r, N). Hence the levelized energy price pe is
Numerical results for this levelizing factor
can be found in Table 14.2.3 for several values of r and re. EXAMPLE 14.2.3
Fuel costs pe = $5/GJ at the start of the first year and grows at a real rate of 4% per year. What is the equivalent levelized fuel price over N = 20 years if the discount rate is r = 6%?
404
Active Solar Collectors and Their Applications TABLE 14.2.3 Levelizing Factor (A/P, r, N)/(A/P, r'e, N) with r'e = (r - re)/ (1 +/•,) for Several Values of Lifetime N, Discount Rate r, and Escalation Rate rc
\
N = 10 c
r(%\/-X /o)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
-1
0
1
2
3
4
0.947 0.948 0.949 0.950 0.950
1.000 1.000 1.000 1.000 1.000
1.115 1.113 1.111 1.110 1.108
.178 1.175 1.172 .170 .167
1.245 1.241 1.237 1.233 1.229
0.951 0.952 0.953 0.953 0.954
1.000 1.000 1.000 1.000 1.000
1.056 1.055 1.054 1.053 1.052 1.052 1.051 1.050 1.049 1.049
1.106 1.105 1.103 1.101 1.100
.164 .162 .159 .156 .154
1.225 1.222 1.218 1.215 1.211
0.955 0.955 0.956 0.957 0.957
1.000 1.000 1.000 1.000 1.000
1.048 1.047 1.046 1.046 1.045
1.098 1.097 1.095 1.094 1.092
.151 .149 .147 .144 .142
1.208 1.204 1.201 1.198 1.195
N = 20
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.904 0.907 0.910 0.913 0.916
1.000 1.000 1.000 1.000 1.000
1.108 1.105 1.101 1.098 1.094
1.231 1.223 1.215 1.208 1.200
1.370 1.357 1.344 1.332 1.319
1.529 1.509 1.490 1.472 1.454
0.919 0.921 0.924 0.926 0.929
1.000 1.000 1.000 1.000 1.000
1.091 1.088 1.085 1.082 1.079
1.193 1.186 1.179 1.173 1.167
1.308 1.296 1.285 1.27 1.264
1.436 1.419 1.403 5 1.388 1.373
0.931 0.933 0.935 0.937 0.939
1.000 1.000 1.000 1.000 1.000
1.076 1.074 1.071 1.069 1.067
1.161 1.155 1.150 1.145 1.140
1.255 1.245 1.236 1.228 1.220
1.359 1.345 1.332 1.320 1.308
SOLUTION
From Table 14.2.3 the levelizing factor for this case is 1.436. Hence the levelized fuel price is $7.18 /GJ. Several features may be noted in Table 14.2.3. First, the levelizing factor is unity if there is no cost escalation. Secondly, for a given energy escalation rate, the levelized cost decreases as the discount rate increases. This makes sense because a high discount rate deemphasizes the role of high costs in the future.
Economic Analysis
405
14.2.4 Discrete and continuous cash flows The formulas derived above assume that all costs and revenues occur in discrete intervals. This is common accounting practice and reflects the fact that bills are paid in discrete installments. Even though a solar energy system produces energy continuously the benefit of this production is realized only in discrete intervals whenever fuel bills are paid. Also the rates for interest, general inflation, fuel escalation, etc., are quoted as annual rates. Hence, we follow common practice and work with the discrete formulas. Nonetheless, it is instructive to consider the continuous case. Whether one chooses a discrete or a continuous analysis is a matter of convention. Instead of an interest rate based on annual compounding, one could quote a slightly lower rate based on continuous compounding; the annual payments would be the same. To establish the connection between the discrete and the continous growth rates, consider something that grows at an annual rate r. At the end of 1 year it will have grown by a factor (1 + r). If, however, the growth is compounded in m equal intervals per year, the growth per interval is (1 + r/m) and by the end of the year the growth amounts to
Using the well known formula
one finds that the annual growth factor corresponding to continuous compounding is If the total growth during the year is to be the same, then the growth rate in the continuous formula must be smaller than the rate in the discrete formulas. Designating the corresponding rates by subscripts and equating the annual growth factors, one finds the following relation between the rates:
For sufficiently small rates, say less than 20%, one can approximate this relation by keeping only the lowest terms in the series expansion of the exponential function
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Active Solar Collectors and Their Applications
With the replacement of rates according to Eq. (14.2.16), the continuous formulas in Table 14.2.4 yield the same results as the discrete ones. To develop a better intuitive understanding of growth rates, it is instructive to consider the doubling time N2 corresponding to a growth rate r. It is given by
or
This is the basis of a very convenient rule, sometimes called the "seventyyear rule": the product of the doubling time and the growth rate r (in units of percent) is 70 years: 70 years (with r = %)
N r = 69.3 ... years
EXAMPLE 14.2.4
If energy costs grow at 5% per year, in how many years will they double? SOLUTION
The doubling time is 70 years 5
= 14 years.
TABLE 14.2.4 Factors for Economic Analysis" Expression for discrete analysis
Expression for continuous analysis
(F/P, r, N)
(1 + r)»
exp (rN)
P
(P/F, r, N)
(1 + r)~N
exp ( — rN)
P
A
(A/P, r, N)
r 1 - (1 + r)~N
exp (r) - 1 1 — exp ( — rN)
A
P
(PI A, r, N)
1 - (1 + /•)-" r
1 - exp(-rJV) exp (r) - 1
Quantity known
Quantity to be found
Factor
P
F
F
a (/( = annual payment, P = present value, /•' = future value, r = discount rate, N — life lime).
Economic Analysis
14.3
407
ECONOMIC EVALUATION CRITERIA
In this section we develop the principal criteria for the economic analysis of a solar energy system. To clarify the issues let us consider the simplest possible case: a solar energy system of capital cost C that delivers an annual amount of useful energy Q, without any expenses for operation and maintenance. (Of course, the formulas apply equally well to an investment in energy conservation.) The system is assumed to last N years without any decline in performance, and any salvage value or benefit after these N years is neglected. The equipment is paid for with cash; this is called equity financing by contrast to debt financing (i.e., borrowing). Once the basic criteria have been explained, it is straightforward to extend the formulas to more complicated cases. In Section 14.4 we add expenses for operation and maintenance and we include the cost of the backup or auxiliary to get the total life cycle cost of the system. Complications arising from debt financing and from taxes are addressed in Section 14.5. 14.3.1 Life cycle savings If one did not install the solar energy system, then the energy Q would have to be supplied by some other energy source, for instance, a gas-fired furnace. Let us designate this alternative as backup or auxiliary. If the price of fuel is pe and if the auxiliary converts this fuel with an efficiency »?aux, then the annual cost of supplying Q would be Qpji\mv This is therefore the value of the annual energy savings due to the solar energy system. If the fuel price escalates at a rate ra one needs to levelize the fuel savings by using the levelized fuel price pe of Eq. (14.2.13): with Hence the levelized annual fuel savings are levelized annual fuel saving = The levelized annual cost of capital is acap,
for system life N and discount rate r. The levelized annual net saving s is the difference between these two terms:
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Active Solar Collectors and Their Applications
The life cycle savings (also known as net present value) are obtained by dividing the levelized annual saving by the capital recovery factor:
This is the true value of the solar energy investment. It is the difference between the life cycle fuel savings and the capital cost. Sometimes one wants to consider the life cycle cost of the entire energy system, not just the solar component by itself. This is described explicitly in later sections. For the moment we merely point out that an analysis based on life cycle cost and one based on life cycle savings are essentially the same, the only difference being the inclusion of the auxiliary. However, the criteria to be discussed next are quite different.
14.3.2 Internal rate of return In order to calculate the life cycle savings one needs to specify the discount rate r. Ideally r should be the interest rate that reflects the cost of the money invested in the solar energy system. However, as discussed in Section 14.2.2, this rate is not always known with sufficient accuracy. Many investors have a wide choice of investment opportunities, each with a different rate of return and a different risk. Such investors like to compare different investments in terms of their individual productivities or rates of return. Hence it is desirable to define a rate of return for a solar energy system in terms of the revenue produced by that system. This rate is called internal rate of return because it is based only on the project itself, without any reference to the cost of money. Suppose an amount of money Cis invested at an interest rate r over N years. Then it produces a levelized annual revenue of
On the other hand, if the amount C is invested in a solar energy system, then it yields a levelized annual revenue
Hence the solar investment behaves like a bank account whose interest rate r is determined by equating the last two equations and solving for n
Economic Analysis
409
This is the internal rate of return of the solar investment, designated by r,. Of course pe is also levelized with discount rate r, as with Comparison with Eq. (14.3.5) shows that the internal rate of return is that discount rate for which the life cycle savings vanish. That is the general definition of the internal rate of return
In general this equation cannot be solved in closed form, but r, can be found by iterations or by using Figs. 14.3.1 or 14.3.2 as discussed below. If r, for a particular project is greater than the average rate of return on other investments of comparable risk, then that project is a good investment; otherwise it is not. Thus the internal rate of return is a rational and a convenient criterion for evaluating the desirability of a solar investment.
14.3.3
Cost of saved energy
Another possible criterion for the evaluation of energy investments is the cost of energy. If a system produces an amount of energy Q and thus saves 2/^aux in fuel per year and if the annual cost for capital is (A/P, r, N)C, then this saved energy costs
Figure 14.3.1 The relation (r,Np)= 1 -exp[-WV,)W Np)] between internal rate of return r,, simple payback period N,,, and investment lifetime N, for continuous analysis.
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Active Solar Collectors and Their Applications
Figure 14.3.2 Relation between internal rate of return, simple payback period, and lifetime of an energy investment for discrete analysis. Shifted axes show effect of energy price escalation re.
The cost psavcd of saved energy has units of $/GJ and is levelized; it can be directly compared with levelized prices of conventional energy forms. The cost of saved energy presumes a discount rate, but it is independent of the fuel escalation rate. When more than one end use energy form is involved (e.g., heat and electricity in a cogeneration system), the definition of the cost of saved energy becomes ambiguous. 14.3.4 Payback period The payback period is denned as ratio of capital cost over annual savings. At the end of the payback period the system has paid for the initial investment and any revenue produced thereafter is pure gain. Several different kinds of payback time can be denned (e.g., simple, discounted, corrected for fuel inflation, etc.), but the simple payback period Np seems to be the most useful.5 It is the ratio of capital cost C and first year savings R = Qpe/Jiam: ^Attempts to overcome the deficiencies of the simple payback period by including discounting, fuel escalation, etc., lead to formulas so complicated one might as well do a proper analysis.
Economic Analysis
41 1
The payback period is a particularly simple criterion but it is not the correct criterion for evaluating energy investments. It fails to account for system life, discount rate, fuel escalation, and benefits accrued during the remainder of the lifetime. It gives a myopic perspective by overemphasizing the short term. It can be very misleading, and economists have been unanimous in rejecting it. Nonetheless it is instructive to think about the payback period. As pointed out by Socolow [1981], the payback period can provide the correct criterion if used in conjunction with the system lifetime. For this discussion it is convenient to use the formulas for continuous compounding. To simplify the argument, let us at first neglect fuel escalation. Let us recall Eq. (14.3.8) for the relation between internal rate of return r, and system cost C:
(with pe as levelized energy cost since pe = pe in the absence of fuel escalation). Dividing by C we obtain the inverse of the payback period on the right-hand side. Using the capital recovery factor for continuous compounding (Table 14.2.4) we can thus rewrite Eq. (14.3.12) in the form
For small rates, say r, < 0.2, one can approximate exp (r,) — 1 by r,. Thus the equation can be rearranged as
This is a one-to-one relationship between the variables y = (r/Np) and x = (N/NP). The dimensionless ratio N/NP is the system lifetime in units of payback period. Equation (14.3.14) is plotted in Fig. 14.3.1, with N/NP as the x axis and r/Np as the y axis. This universal curve
contains much wisdom and is worth knowing. First, when a system lasts forever, x -» oo, then y -» 1; i.e., the rate of
412
Active Solar Collectors and Their Applications
return is one over the payback period
Second, the return on investment r, is positive if and only if the system life is longer than the payback period, quite an obvious result. Thirdly, the figure shows that y increases rapidly between x = 1 and x = 2, reaching 0.8 at x = 2.0. Beyond x = 2 it grows much more slowly. This means that the rate of return r, is very sensitive to the value ofN/Np as long as N/Np is small, say less than 2. But beyond that range, further increases in lifetime bring only a relatively small benefit for the rate of return. For example, at N/NP = 3 the rate reaches 94% of its asymptotic value l/Np, only a small increase over the value Q.8NP at N/NP = 2. Hence system life is critical for N < 2NP, but for N > 3NP the rate of return is close to its asymptotic value. This analysis is readily generalized to include the effects of fuel price escalation or of operating costs. If fuel prices escalate at a rate /•„ then pe in Eq. (14.3.12) is to be replaced by the levelized price of Eq. (14.2.13), with the formulas for continuous compounding of course,
Repeating the argument leading to Eq. (14.3.14), one obtains Eq. (14.3.14) with only one modification: rt is replaced by r, — re. Hence the curve in Fig. 14.3.1 still applies if the y axis is interpreted as the difference between the internal rate of return r, and the fuel escalation rate re.6 Without the approximation exp (r,) — 1 ^ r, used for Eq. (14.3.14), the relationship depends on r, and on Np separately. Hence one needs a family of curves. To derive these curves for arbitrary fuel escalation let us rewrite Eq. (14.3.8) explicitly in terms of first year fuel price as
Inserting the payback period one obtains the result
This relationship is plotted in Fig. 14.3.2, based on the discrete capital recovery factor of Eq. (14.2.6). The curves are labeled by the system life N ""Operating and maintenance costs (if they can be assumed to grow like fuel costs) can be included by replacing 2p,/i)aux by Qpe/rja^ — fa&MC in the equation for the payback period.
Economic Analysis
413
and, for a given payback time Np on the abscissa, one finds the corresponding value of fe on the ordinate. If the energy price escalation rate re is zero, then fe is equal to the internal rate of return r,. Otherwise it is given by
To provide a nomogram for finding r,, lines have been drawn parallel to the ordinate corresponding to re = 1%, 2%, 3%, and 4%.
EXAMPLE 14.3.1
A solar domestic water heater costs C = $1000 and supplies Q = 12 GJ per year. The backup has an efficiency ?jaux = 0.70 and burns oil costing pe = $7/GJ. The oil price remains constant over the system life of N = 20 years. Calculate (a) the payback period (b) the life cycle savings at r = 5% discount rate (c) the cost of saved energy at r — 5% (c) the internal rate of return SOLUTION
(a) Equation (14.3.11) yields the payback time as
(b) The capital recovery factor is
and the life cycle savings are, from Eq. (14.3.5),
(c) The cost of saved energy is obtained from Eq. (14.3.10),
at r = 0.05. (d) The internal rate of return can be found from Fig. 14.3.2 or, by trial and error, from Eq. (14.3.8), and turns out to be
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Active Solar Collectors and Their Applications
EXAMPLE 14.3.2
Same as Example 14.3.1 but with oil price escalating at re = 4% per year. SOLUTION
The levelized oil price is
and the life cycle savings are
For the internal rate of return the oil price must be levelized with discount rate rn as in Eq. (14.3.18). The result for the rate of return is
The price of saved energy and the payback time remain the same as before.
EXAMPLE 14.3.3
Same as Example 14.3.1 but performance degrades by 1% per year while oil price stays constant. SOLUTION
Now we must levelize the energy Q according to Eq. (14.2.13): with where rQ is the yearly loss of performance; in our case rQ = —0.01. The result is (from Table 14.2.3 with N = 20, r = 5%, and re = -1%)
With that we obtain the new life cycle savings
much lower than before. For the internal rate of return calculation Q is of course levelized using r, as discount rate, analogous to Eq. (14.3.18) (with r'Q instead of r£). The result is
415
Economic Analysis
The levelized cost of saved energy increases to
because less energy is saved in later years. The payback period remains, of course, unaffected. 14.3.5 Comparison of criteria We have presented four basic criteria for the evaluation of energy investments. Their main characteristics are summarized in Table 14.3.1. Which criterion one should use depends on the circumstances. In particular, it depends on the assumptions one is willing to make. Ideally one would like to calculate life cycle costs or life cycle savings. As explained in Section 15.1 the life cycle cost (or equivalently the life cycle saving) is the appropriate criterion for system optimization. However, one can calculate life cycle costs or savings only if one specifies the discount rate and the energy escalation rate. If one does not know the discount rate but is willing to make a guess about energy escalation, one can calculate the rate of return based on the annual revenue produced by the system. This rate is called internal rate of return and provides a comparison with all types of investments independent of their magnitude. The internal rate of return is that discount rate for which the life cycle savings of the system vanish, or equivalently for which the cost of solar energy equals the cost of backup energy.7 Conversely, if one 'For a + + + . . . . . . + + + annual cash flow sequence, the equation for r, may have two solutions, and r, is not a suitable criterion. This can happen if the debt fraction is high.
TABLE 14.3.1 Comparison of Economic Evaluation Criteria Criterion
Input
Comments
Life cycle saving = Ciifei no S0|ar — Qfe, with soiar
Discount rate and energy escalation rate
This is the true saving. Appropriate for optimization of system parameters. No direct comparison of profitability with other investments.
Internal rate of return
Energy escalation rate
Internal rate of return r> is value of discount rate for which life cycle savings vanish. Provides direct comparison with profitability of other investments.
Cost of saved energy
Discount rate
Only good for ranking of energy investments that involve a single end use energy form.
Payback period
Very simple, but misleading if taken by itself. However, in conjunction with system lifetime it yields internal rate of return directly from Fig. 14.3.1 or Fig. 14.3.2.
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Active Solar Collectors and Their Applications
knows the discount rate but does not want to guess the future energy prices, one can still rank energy investments in terms of their cost of saved energy. Finally there is the payback period, a simple criterion but much maligned. However, when combined with the system lifetime, the payback period yields directly the internal rate of return, at least for the simple case considered so far. Sometimes it is desirable to have an index that characterizes the economics of a solar installation without any reference to a specific economic scenario. A suitable and frequently used index is the ratio of capital cost Cand annual delivered energy Q. It is sometimes called the energy capacity cost:
energy capacity cost For a given annual charge rate Jcr [see Eq. (14.4.9)], this quantity yields directly the cost of solar energy [in $/GJ] as
If in addition the cost of operation and maintenance is known, one can calculate the internal rate of return. Hence the energy capacity cost is a very convenient criterion for ranking solar installations. Another useful criterion is the saving/investment ratio SMc/C. 14.4 OTHER COSTS So far we have restricted our attention to the solar components of the system. A typical solar energy system contains a backup or auxiliary as well. Frequently one wants to know the total cost of supplying the energy requirements of the load. The total capital cost Csys of the system is the sum of the solar cost C and the cost Caux of the backup:
Almost all solar systems have a complete backup that is determined by the load alone, independent of the solar components. The total annual load is Qtot. A portion Q
of this is supplied by the sun. The levelized annual cost for the auxiliary is
Economic A nalysis
417
and the levelized annual cost of capital is
The total annual cost is the sum of these two terms
The life cycle cost Clife is obtained by dividing the levelized annual cost by the capital recovery factor
Without solar the cost would be
Comparison with Eq. (14.3.5) shows that the life cycle savings of the solar portion are the difference between the life cycle cost without solar and the life cycle cost with solar
Since C|ife no so,ar is independent of the solar portion we see that the life cycle savings of the solar portion are a constant term minus the total life cycle cost. Therefore a life cycle cost analysis is essentially equivalent to an analysis of life cycle savings. In particular for system optimization there is no difference between maximizing Stife or minimizing C|ife, since the derivative of a constant vanishes. Most mechanical systems require some maintenance. In addition there will usually be operating expenses; e.g., to power the pump(s). Thus the analysis would be incomplete without consideration of these costs. The first year cost a0&M for operation and maintenance may of course change, say at an annual rate r0&M, in which case it needs to be levelized according to Eq. (14.2.14):
with
41 8
Active Solar Collectors and Their Applications
Including operation and maintenance the expression for the life cycle cost becomes
Sometimes the expense for operation and maintenance is expressed as a fraction 7^&M of the capital cost. In that case the total levelized annual cost can be written in the form
The factor multiplying the system cost Csys is sometimes called annual charge ratejm
For the sake of completeness we restate the formulas derived in the preceding section for life cycle savings, cost of saved energy, and payback time if there are expenses for operation and maintenance. In that case the annual energy cost savings QpJ^^ must be reduced byJ^&MC. The resulting expression for the life cycle savings is
The internal rate of return r, is of course still determined as that value of the discount r for which Sm vanishes. The cost of saved energy is
Finally the payback time becomes
14.5 DEBT FINANCING, TAXES, AND INFLATING CURRENCY The formulas derived in the preceding section do not include any effects of tax deductions. They are appropriate for investments made by nonprofit organizations or by homeowners who pay with their savings. They should also be used when assessing the benefits of a technology for society. On the other hand, homeowners who borrow money can deduct the interest pay-
Economic Analysis
419
ments from their income tax. For profit-making organizations the analysis is even more complicated because an energy investment can be depreciated and because the cost of fuel is deductible. 14.5.1
Principal and interest
Part of the annual payment A for a loan is interest while the rest, called principal, serves to reduce the loan balance. U.S. tax law allows interest payments as a tax deduction, while payments for the principal are not deductible. Therefore a tax-paying investor needs to know what portion of the loan payment A is interest. Let /„ = interest and pn = principal in year n. The sum is equal to A:
During the nth year the remaining debt is
Hence the interest for the «th year is
Comparing in+l with in one finds
By means of Eq. (14.5.1) pn can be eliminated, with the result
This recursion relation has the solution
as can readily be proved by mathematical induction. This can be rewritten in the form
It is worth noting that the year n enters Eq. (14.5.7) only in the combination (n — N). This implies that the fractional allocation to principal and interest depends only on the number of years (n — N) left in the loan, not on the original loan period. A loan has no memory, so to speak. In general the loan interest rate r, is different from the discount rate r used
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Active Solar Collectors and Their Applications
in the economic analysis. Furthermore, the loan period NI may be different from the system life N. Hence these rates and times have to be kept apart when calculating the present value of the tax deduction for interest. In Eq. (14.5.7) the interest payment /„ in year n for a loan amount Pioan is given as
The present value PiM of all interest payments is found by discounting each /„ by (1 + r)~" and summing over n
The result can be written in the form
with rj = (r — r/)/( 1 + r,). If the incremental tax rate is T, the total tax payments are reduced by rPml. [This assumes a constant tax rate; a variable tax rate would have to be included in the summation in Eq. (14.5.9).] EXAMPLE 14.5.1
A homeowner finances a $2000 solar water heater with a 5-year loan at r, = 8% interest rate. His discount rate r is also equal to 8%, and his income tax bracket is r = 40%. The levelized annual fuel savings (including O&M) are R = $200. What are the life cycle savings of this water heater if it lasts N = 20 years? SOLUTION
First find the present value P-ml of the interest using Eq. (14.5.10) with r1, = 0, (A/P, fh N,) = I/TV, = 0.20 and (A/P, r,, N,) = 0.2505 = (A/P, r, N,)
The life cycle fuel savings are
The total life cycle savings are obtained by adding r.Pint to Eq. (14.3.5):
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Economic Analysis
(In this particular example the savings are positive with tax deduction, negative without.) 74.5.2
Depreciation
U.S. tax law allows business property to be depreciated. This means that for tax purposes the value of the property is assumed to decrease by a certain amount each year and this decrease is treated as a tax deductible business loss. For an economic analysis it is convenient to express the depreciation claimed over the year as an equivalent present value. The tax law may permit several different schedules for depreciating property. For straight line depreciation over JVdep years the present value of the total depreciation is, as fraction of system cost,
For sum-of-yearly-digits depreciation the corresponding formula is
[Dickinson and Brown, 1979]. (With changes in U.S. tax law, the formula for/iep has changed). 14.5.3 The complete formula The government may choose to stimulate the commercialization of a new technology by granting tax credits. This is current U.S. policy vis-a-vis conservation and renewable resources. A tax credit rcred on an investment / means that the tax payments are reduced by Tcred/. For example, up to $10,000 of the cost of a residential solar system qualifies for a 40% federal tax credit. Thus a homeowner buying a $ 3000 solar water heater saves $ 1 20 in income tax, and in effect the water heater costs him only $1800, provided that his tax liability exceeds his tax credit. Some states grant state income tax credit in addition to any applicable federal credit. (The U.S. solar tax credits expired at the end of 1 984.) Taking into account all applicable tax deductions one obtains the following formula for the life cycle cost of a solar energy system for a homeowner: cost of auxiliary energy down payment
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Active Solar Collectors and Their Applications cost of loan tax deduction for interest payments tax credit operation & maintenance salvage value
where
Csys = total system cost = C + Caux ./tan = fraction of investment paid by loan ./O&M = levelized annual cost of operation & maintenance, as fraction of Csys /s"alv = salvage value, as fraction of Csys / = general inflation rate TV = system lifetime TV/ = loan period pe = first year energy price Qaux = Got ~ Q — annual auxiliary energy r(. = market energy escalation rate r'c = (r- fe)l(\ + re)
r - market discount rate r, = market loan interest rate r', Tcred T ??ailx
= = = =
(r-./•/)/(! + r,) tax credit incremental income tax rate fuel efficiency of backup
For an industrial investor the formula is different because fuel costs and operating costs are tax deductible and the cost of the solar energy system can be depreciated. The equation can be stated either in terms of before-tax cost or in terms of after-tax cost. These two forms of the equation differ only by an overall factor of (1 — T), where T is the income tax rate. For example, if the market price of fuel oil is $6/GJ and T - 50%, then the before-tax cost of fuel is $6/GJ while the after-tax cost of fuel is (1 - T) X $6/GJ = $3/ GJ. Stated in terms of after-tax cost the equation for the life cycle cost of an
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Economic Analysis industrial solar energy system reads cost of auxiliary energy down payment cost of loan tax deduction for interest payments tax credit depreciation operation & maintenance salvage
The symbols are the same as in the previous equation. fdep is the present value of the depreciation given by Eq. (14.5.11) or (14.5.12), and Tsa,v is the salvage tax rate, which may be different from the corporate income tax rate. For simplicity f0&M is assumed to include property tax and insurance, as well as the levelized cost of repairs and overhauls. The rate r is the after-tax discount rate; in other words when setting the life cycle savings to zero, one obtains the after-tax internal rate of return. 14.5.4
Constant currency versus inflating currency
To close this section let us examine under what conditions an analysis in constant currency is equivalent to one in inflating currency. For an equity investment without tax deduction the life cycle cost is simply
if we neglect for simplicity O&M and salvage. Inflation can enter only through the rate fn
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Active Solar Collectors and Their Applications
that accounts for energy escalation [Eq. (14.2.11)]. Suppose r and re are the market discount and the market energy escalation rates, respectively, while the general inflation rate is /'. Then the real rates are
Inserting r = i + (1 + i)rrcat and re — i + (1 + i)re real into Eq. (14.5.16) one obtains
This rate is the same whether calculated with real or with market rates! The inflation rate drops out completely. This proves the claim made earlier that a constant currency analysis is exactly equivalent to one in inflating currency for an equity investment in the absence of taxes. This conclusion is not changed if operation and maintenance, salvage, and tax credit are included. On the other hand, for loan payments and tax deductions inflation can have a real effect. One can see this when one tries to rewrite the loan payments in terms real rates and finds that, for example,
Numerically the effect is small but quite noticeable for inflation rates around 5%-10%. Dickinson and Brown [ 1979] present some examples. For an inflation rate / = 6% they find that there can be a real difference on the order of 10% between a constant dollar analysis and one in inflating dollars. But if both depreciation and loan terms are present, the errors tend to compensate and the net error is only a few percent. For an intuitive explanation of the difference between a constant dollar and an inflating dollar analysis, consider the deduction for straight-line depreciation. Since the depreciation allowance has a fixed numerical value in inflating dollars, its real value in future years is decreased by inflation. In the limit of very high inflation the investor loses the benefits of all but the first year's depreciation. REFERENCES Dickinson, W. C. and Brown, K. C. 1979. "Economic Analysis for Solar Industrial Process Heat Systems." Report UCRL—52814. Livermore, CA: Lawrence Livermore Laboratory.
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Jones, B. W. 1982. Inflation in Engineering Economic Analysis. New York: Wiley Interscience. Kreider, J. F. 1979. Medium and High Temperature Solar Processes. New York: Academic Press. Riggs, J. L. 1982. Engineering Economics. New York: McGraw-Hill. Socolow, R. H. 1981. "The Utility of the Payback Period in Scaling the Durability of Savings." Internal note. Princeton, NJ: Center for Energy and Environmental Studies.
15. SYSTEM OPTIMIZATION
Nowadays people know the price of everything and the value of nothing. Oscar Wilde
Having developed the formulas for evaluating system performance and cost, one can address the question of system optimization. In a narrow sense this means varying the system configuration and the sizes of the system components until the most cost-effective combination has been found. In a broader sense one would like to identify the best match between solar technologies and the energy needs of society, taking into account the full social costs. 15.1 OPTIMIZATION CRITERIA In Chapter 14 several criteria were developed for evaluating the economic attractiveness of an energy investment; in particular, life cycle cost (or savings), internal rate of return, cost of saved energy, and payback period. The system optimization depends on which criterion one chooses to measure the economic benefit, and the reader may wonder which criterion to choose. Minimization of total life cycle costs is the rational choice, for the following reason. Supplying energy is not a goal in itself but only a necessary step in order to provide comfort or produce goods. To maximize the benefit of these goods, one needs to minimize the cost of providing them. Hence one should minimize the life cycle cost of an energy system (if, as is usually the case, it is appropriate to consider the energy system in isolation from other items). By contrast, optimization according to an internal rate of return or cost of saved energy amounts to an inappropriate suboptimization as can be seen from the following example. Suppose a factory needs to heat water from the water mains to 70°C at a constant flow rate and assume collector costs proportional to area. Since the temperature of the water mains is close to the average ambient temperature, one could use a small flat plate collector field as preheater to raise the water temperature by just a few degrees. Since the collector efficiency would be very high, the cost of this solar energy is low; equivalently the internal rate of return for this small collector field is high. 426
System Optimization
427
If the collectors are cost effective even when operating at 70°C, then one can further reduce the total life cycle energy costs by making the field large enough to heat the water to 70°C, even though the extra collectors operate at lower efficiency. For these incremental collectors the cost of solar energy is higher and the internal rate of return lower than for the small collector field. Maximization of life cycle savings corresponds to a large collector field while maximization of rate of return would yield a small collector field and small savings. This example demonstrates that the internal rate of return, while a valuable tool for comparing the attractiveness of energy investments, is not the correct criterion for system optimization.
15.2
OPTIMIZATION
15.2.1 Equations for optimization The life cycle cost C and the delivered solar energy Q are functions of the system components; in particular, collector area, storage capacity, heat exchanger size, flow rate, etc. Suppose there are n such variables x}, x2, . . . , xn, to be optimized. This means that one wants to vary these n variables until one finds the minimum of the life cycle cost,1 Eq. (14.4.3),
where (2tot = total annual demand for delivered energy Pe/T]m* - effective energy cost [ = market price, levelized according to Eq. (14.2.13) and divided by efficiency i?aux of backup] (A/P, r, N) = capital recovery factor, Eq. (14.2.6) Cut = C + Caux = total capital cost of system (solar + backup) The life cycle cost is minimized by setting the derivatives of Ciife with respect to all the x, equal to zero. Since only Q and C,0, depend on x,, this implies the system of n equations
'To keep the argument simple, we neglect operation and maintenance as well as possible effects of taxes. In a real calculation they should certainly be included and therefore one should use the full formulas, Eq. (14.5.13) or (14.5.14) instead of (15.2.1). If only operation and maintenance need to be added, then the modification of Eqs. (15.2.1) to (15.2.5) is particularly simple: replace (A/P, r, N) by (A/P, r, N) + J0&u where J0&M is the levelized annual expense for operation and maintenance, expressed as a fraction of the capital cost of the system.
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Active Solar Collectors and Their Applications
These equations have a simple intuitive interpretation. At the optimum for each variable the incremental cost
of increasing the system equals the incremental value in energy savings
(This is a general optimization rule in the entire field of economics.) Usually one will be able to choose among several different manufacturers and models for the system components. In that case the optimization is to be repeated for each possible (and reasonable) combination of models to identify the best overall choice. As an important illustration consider the case where only the collector area A and the storage capacity M are to be optimized, and where costs vary linearly with A and with M,2
where c0 = fixed cost CA = incremental collector cost CM = incremental storage cost Then Eq. (15.2.2) becomes
and
If Q is given as an explicit function of A and M, these two equations can be readily solved to find the optimal values A0ft and Mopt. Otherwise one will resort to a numerical search of the results of computer simulations. 2 The linearity of system cost with collector area seems to be quite a good rule. Storage costs are more complicated, being determined not only by the storage material, but by the container cost (approximately proportional to A/2'3), and by the charging and discharging rates.
System Optimization
429
The optimization is particularly simple for systems without storage. This is illustrated in the following paragraphs for photovoltaics and for process heat. 15.2.2 Photovoltaics with sellback to utility Storage of electricity is quite expensive, and for customers that are connected to the utility grid it may be more cost effective to rely on the utility for backup than to store energy on-site. Thus in a wider sense the utility can provide storage, by averaging over many customers and by using the energy storage implicit in its fuel supply. In some cases the characteristics of the utility (its power plant types and demand profile) will make storage of electricity desirable. But even then it is likely that the utility can store electricity at a lower cost than local storage at the customer's side because of economies of scale and because the utility can average over demand fluctuations of the individual customers. A crucial feature of this grid integration is the possibility of selling excess electricity (i.e., greater than the demand of the local customer) back to the utility. There is a large potential for local power generation (a lot of free surface area is available on roofs of buildings), but this potential will be utilized only if the utility pays an adequate price for excess electricity. In the United States this has been recognized and codified by the law regulating public utilities [PURPA, 1978]. This law requires the utility to pay a fair price, essentially the avoided cost, to anyone who wants to sell electricity. The avoided cost is the cost that the utility would otherwise have to pay in order to generate this electricity. For example, if at a particular time it would cost a utility $20 to generate an additional GJ, then the utility has to pay $20 to anyone who is willing to sell a GJ to the utility. The PURPA regulation stimulates competition and enhances the overall economic efficiency of electric power generation because it encourages local generation whenever that can be done at a lower cost than the cost of central generation. A regulation of this type is a boon for many alternative energy technologies because their optimal scale is small. An exact optimization of the integration of local photovoltaic generators into the grid of a central utility is complicated and highly dependent on the characteristics of the utility. But from the perspective of the local customer it is only the rate structure of the utility that matters. The charges for electricity may depend not only on the total quantity but on time of day and on instantaneous demand. In the following we discuss as an important example the case of a user with constant daytime demand L, as is typical of industrial operations. For this case the optimization can be carried out in closed form. For photovoltaic systems the generated electricity is proportional to collector area A because the efficiency r\e is essentially constant and independent of A. Thus the optimization condition, Eq. (15.2.4), will generally not have a solution. If the cells are cost effective at all, then the savings increase linearly with A, up to the point where electricity is generated in excess of the
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load. At that point there is a break in the value of the electricity. Let us designate the purchase price of electricity by #,uy and the selling price (to the utility) by p^. (The case where the excess must be discarded corresponds to Aeii = 0-) To find the optimal area Aopl we need to rederive the equation for the life cycle savings. The instantaneous electric output is
where / is the instantaneous irradiance on the collector. Selling (or storing or dumping) becomes necessary whenever this output Q exceeds the load L, which is assumed to be constant. As discussed in Section 11.3.4 one can define an insolation threshold X for sellback; excess is generated whenever
Of course, this can occur only if A is larger than the threshold area Axn,
Both the annual total electricity produced by the cells and the excess above the threshold can be calculated with the help of the correlation q(X) ofEq. (11.3.2):
The total production is
Of this a portion Qxlb
is above the threshold for selling and can be sold at the price psc,r; the rest,
displaces electricity that would otherwise be purchased at the price pbur Hence the levelized value of the annual electricity production is
System Optimization
431
The life cycle savings Sti{e are obtained by dividing by the capital recovery factor (A/P, r, N) and subtracting the initial capital investment (c0 + c^A):
With Eqs. (15.2.10) and (15.2.11) this can be written as
Let us now find the optimal area Aopt that maximizes S\ife. In most cases Aeii is less than pbuy, and we shall assume that to be the case (otherwise one would sell the entire production and there would be in effect only one price). If the system is cost effective even at the lower electricity price psdb then again there is no optimum; rather one would make the area as large as permitted by other constraints such as land availability. More likely is the case where psen is relatively low and Eq. (15.2.13) has a genuine optimum obtained by setting the derivative with respect to A equal to zero:
Inserting the polynomial (15.2.9) for q(K) and Eq. (15.2.7) for Zone obtains after some straightforward algebra the result
EXAMPLE 15.2.1
A factory with constant electric load L — 10 kW is in a location with latitude X = 32° and yearly average daytime beam irradiance //, = 0.523 kW/ m2. The levelized electricity prices are pbuy = $20 /GJ and pM = $10 / GJ($10 /GJ = 3.6 0/kWh). A photovoltaic system with r,e = 10% efficiency can be installed for CA — $100/m2 (this corresponds to cA/(ri,, X 1 kWpk/m2) = $ 1 /Wpk in terms of the dollar per peak watt rating); the fixed cost c0 = 0. Evaluate the life cycle savings without sellback and at the optimal area, for discount rate r = 8% and system life TV = 20 years. SOLUTION
The capital recovery factor is (A/P, r, N) = 0.1019. Evaluating the curve fit listed under Fig. 11.3.1, one finds the coefficients of the polynomial q(x) to
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Active Solar Collectors and Their Applications
be
q0 = 7.384 GJ/m2, ql = n^OCGJ/m^kW/m 2 )- 1 q2 = 6.046 (GJ/m2)(kW/m2r2 The peak insolation /max for this location is the value of X for which q(X) vanishes; solving the quadratic equation
one finds (choosing the smaller solution, of course, because the curve fit is not good past the first zero)
The largest area without sellback is given by Eq. (15.2.8),
At this area the life cycle savings are
It is positive, and hence the optimal area is larger than A^. The optimal area is found from Eq. (15.2.15):
The corresponding threshold is
System Optimization
433
Inserting q(Xopt) = 1.06 GJ/m2 into Eq. (15.2.13) one finds the maximum of the life cycle savings,
EXAMPLE 15.2.2
Same as Example 15.2.1 but higher electricity prices pbuy = $25/GJ and Aell = $15/GJ. SOLUTION
Now the denominator in Eq. (15.2.15) is negative and there is no real solution for Aopt. This is a reflection of the fact that the system would be cost effective even if all the electricity were worth only $15/GJ; hence the savings increase indefinitely with A. Another way to see this is to note that the cost of saved energy, Eq. (14.3.10) (with ?jaux = 1.0 for electricity, and with C = c^A and Q - q^Ai]^ is
if no energy is discarded. Since pxK is greater than psaved in this example, it is profitable to generate as much as possible. Note that the denominator vanishes as /7sdi approaches psaved, and in that limit Aopl becomes infinite.
15.2.3 Industrial process heat systems Industrial process heat systems with constant load rate L are another instance where the optimization can be carried out in closed form [Gordon and Rabl, 1982]. For systems without storage the basic equations have been derived in Section 12.3. The features of the optimization are quite similar to the photovoltaic example. For simplicity let us consider only direct systems; i.e., set Fx = 1. A critical quantity is the threshold ^dump for dumping of energy in excess of the load (unlike photovoltaics, selling of excess heat will rarely be worthwhile). Adumf is given by Eq. (12.3.10):
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Active Solar Collectors and Their Applications
The annual useful energy Q is given by Eq. (12.3.5) if A is less than J dump and by Eq. (12.3.20) if A is greater:
This must be reduced if the system does not operate every day. The notation is as in Section 12.3; in particular, L is the load rate and q(X) = q0 — q\X + q2X2 is the correlation of Eq. (1 1.3.2). This Q is inserted into the equation for life cycle savings, Eq. (14.3.5),
Setting the derivative with respect to A equal to 0, one obtains Eq. (15.2.4) for the optimal area Aopl,
This can be solved since Q is given explicitly as illustrated by the following example. EXAMPLE 15.2.3
A factory at a location with latitude 32° and Ih - 0.523 kW/m 2 needs to heat water from the mains (at temperature Tm = Tamb = 19°C) to the process temperature r]0ad = 80°C. The load is constant at L = 191.5 kW, corresponding to a constant water flow rate w,oad = 0.75 kg/sec. The process runs 24 h/day, 290 days of the year, the rest being downtime for holidays, weekends, and maintenance. The levelized effective fuel cost of the backup is pj T?aux = $16.88/GJ. The solar system costs CA = $165 /m2 (with c0 = 0), and operation and maintenance costs JO&M = 2% of the capital cost per year. The collector parameters are Fmij0 = 0.74 and FmU = 5 W/m2 K. Calculate the useful yearly energy Q as a function of area and find the optimal area that maximizes the life cycle savings, assuming a discount rate r = 5% and a system life N = 10 years. SOLUTION
The polynomial q(X) of Fig. 11.3.1 has the coefficients
System Optimization
435
0, = n^OCGJ/m^kW/m 2 )-', q2 = 6.046 (GJ/m2)(kW/m2;r2. This implies a peak insolation Imax = 1.026 kW/M2,
corresponding to q(X = 7max) = 0. The threshold area for the onset of dumping is obtained from Eq. (15.2.16): Adump = 334 m2.
Evaluating Eq. (15.2.17) one finds
where the factor 290/365 has been added to account for the number of days of actual operation. Q and life cycle savings are plotted in Fig. 15.2.1. The life cycle savings increase from $82,300 at A = AAump to a maximum of $84,600 at the optimal area Aopl = 397 m2. The optimum is really very broad, and Aopl need not be determined with great accuracy. The figure may be misleading in this regard unless one keeps in mind that the origin has been shifted in order to focus on the region of interest. Also shown in Fig. 15.2.1 is the internal rate of return r-t as a function of A. (Note that rt depends on the fuel cost escalation rate; for Fig. 15.2.1 an escalation rate of 10% per year over N = 10 years has been assumed.) Figure 15.2.1 exemplifies the point made in Section 15.1 about the choice
Figure 15.2.1 Annual useful energy Q, life cycle savings (L.C.S.), and internal rate of return r, as functions of collector area A. The specific numbers are based on Example 15.2.3 (From Gordon and Rabl [ 1982]).
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of the optimization criterion. The area that maximizes life cycle savings is larger than the one that maximizes rate of return. The latter drops sharply as soon as dumping sets in. 15.3
SYSTEM VALUE
15.3.1 Definition of value The value V of a solar energy system is defined as its highest permissible cost; i.e., the highest cost at which it can compete with conventional energy. Suppose the solar energy system has a capital cost Cand delivers an amount Q of useful solar energy per year. It saves conventional energy, for example, fuel with a levelized price /7fud that is burned in a boiler with efficiency ?jaux. Then the effective cost of conventional energy is
The annual levelized cost of the solar energy system is JaC, where fcr is the annual charge rate [see Eq. (14.4.9)] for capital and for operation and maintenance (for an equity investment without taxes the charge rate is
where (A/P, r, N) is the capital recovery factor, and JOSM is the levelized annual cost for operation and maintenance, expressed as fraction of the system cost C). The system is cost effective if and only if the annual cost is less than the annual fuel savings, in other words if
The value V of the solar energy system is therefore given by
It depends only on this ratio, not on the individual factors. Also, the value depends equally on performance and on the assumed economic scenario. EXAMPLE 15.3.1
A system delivers 10 GJ per year and displaces natural gas with a levelized price /?fuci = $7/GJ, the efficiency of the backup being r;aux = 0.70. The capital recovery factor is 0.08 (corresponding, for example, to a discount rate r = 5% and a system life TV = 20 years) and the annual charge for operation and maintenance is f0&M = 2%.
System Optimization
437
SOLUTION
The annual charge rate isfcr = 0.10, the effective fuel cost is $7/GJ/0.70 = $10/GJ, and hence the value of the system is
If the system costs more, it is not cost effective under this particular economic scenario. 15.3.2
Value of system changes
Frequently the question arises whether a system could be made more cost effective by a particular modification. For example, one might replace ordinary glazing by more expensive glazing with antireflection coating to improve the performance. Or one might use steel absorber plates instead of copper, reducing both cost and performance. As the following discussion shows, there are two basic criteria that must be satisfied if a system change is to be cost effective. Let us start with a reference system ("old") which costs Cold dollars and delivers QM per year. The value Fold of the system is defined as the highest permissible cost of a cost effective system
Now let us consider a change in the system which alters the cost by
and the performance by
Under what conditions is this change cost effective? Clearly the new system must be cost effective,
otherwise one would not build it. But in addition, one must examine the cost effectiveness of the change itself. Only if
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is the change worth the trouble; after all, the investment for the change must be judged by the same criteria as any other solar energy investment. The value Fchange of the system change is the highest permissible cost of the system change. The two inequalities, Eqs. (15.3.8) and (15.3.9), imply that Fchange is determined by two conditions. The highest permissible cost of the new system is the sum of C0]d and Fchange, and of course it must satisfy Eq. (15.3.8):
In addition Fchangc is bounded by the right-hand side of Eq. (15.3.9).
Change is the lowest number consistent with these two inequalities, and it can be written in the form
The fact that the value of a system change is bounded by two conditions, rather than a single one, has frequently been overlooked. Which of the two conditions is more restrictive depends on the circumstances, as illustrated by the following example. EXAMPLE 15.3.2
Suppose the reference system costs CM - $1100 and delivers Qold = 10 GJ per year. If the effective fuel price is $10/GJ and the annual charge rate/J r = 0.10, then the value is
less than the system cost; i.e., it is not worth building. Consider now a system change that increases the annual energy output by A(2 = 2 GJ to Qnew = 12 GJ per year. Then the value of the system change is
Had one looked only at Eq. (15.3.9), one would have reached the erroneous conclusion that the value of the change would be $200. The latter is not correct because the old system is not cost effective, and the cost of the change must be low enough to make the new system competitive.
System Optimization
439
EXAMPLE 15.3.3
Assume the same conditions as in the preceding example, except for a lower cost of the reference system Cold = $900. In that case the value of the change is
because the old system by itself is already competitive. 15.4 VALUE OF STORAGE 15.4.1
General formulation
To assess the value of storage for solar energy systems, it is instructive to think of the addition of storage as a change in the system that affects both cost and performance. In this sense the addition of storage is like any other change in a solar energy system, for example, like the replacement of nonselective by selective coatings in solar collectors. To compare a system with and without storage, let us use subscripts "ns" for "no storage" and "ws" for "with storage." In general the cost, collector area, and performance of these two systems will be quite different. A nostorage system will usually be small and supply only a portion of the daytime load, while a system with storage will be larger and can supply a larger portion of the load.3 The cost of the no-storage system is
where c0 = fixed cost and CA = cost per collector area [in $/m2], while the system with storage costs
where Cstor is the cost of storage. The incremental cost AC = Cws — Cns is therefore
and it must be less than the incremental value of the fuel savings
-'While storage is essential for space heating, because of the mismatch between the times of supply and demand, the situation is different for power generation and process heat because there is a large demand during daylight hours; in this situation a system without storage may be able to provide solar energy at a lower cost per unit energy than a system with storage.
440
Active Solar Collectors and Their Applications
in order to satisfy criterion (15.3.9). The other criterion, Eq. (15.3.8), becomes
The value Fstor of storage is defined as the highest permissible cost of storage consistent with Eqs. (15.4.4) and (15.4.5). Isolating the storage cost on the left-hand sides of Eqs. (15.4.4) and (15.4.5), one can write the equation for the storage value Fstor in the form
The second term in Eq. (15.4.6) increases with gws — <2ns. Roughly speaking (2ws — Qns is the yearly amount of energy that is cycled through storage. For a given storage capacity (?ws — Qns increases with the number of times that storage is charged and discharged. This is the basis for the folklore that the value of storage is proportional to the number of times storage is used. It explains why, for a given storage technology, long term storage is less cost effective than short term storage. In particular, seasonal and annual energy storage of heat can be practical only with the cheapest materials (i.e., water, solar ponds, aquifers) and the like. However, the simple folklore is not exact, and for a careful analysis one should use Eq. (15.4.6). In fact, some of the examples in the next subsection show that sometimes the value of storage does not increase with (?ws — Qns because of the second criterion in Eq. (15.4.6). A general feature of Eq. (15.4.6) is the dependence of storage value on collector cost. This fact may appear disturbing, since it precludes the possibility of assigning an intrinsic value to storage. This is inescapable. Storage is, by its very nature, a dependent technology: first a product must be supplied, then it can be stored. For example, who would build a wine cellar, if wine were not worth making? This remark is not to belittle the importance of storage. In fact, the very stability of our society rests on our ability to store goods for future use. Storage of solar energy is crucial if solar is to make a large contribution to our energy demand. In the near term, however, while solar equipment is relatively expensive, it is worth looking at applications where storage of solar energy is not essential, because such applications may be more cost effective. 15.4.2
Value of storage for industrial process heat
It is instructive to apply the formalism of the preceding subsection to the case of storage of solar industrial process heat. There are two types of storage
441
System Optimization
to be distinguished: storage from day to night, and storage of energy collected during weekends without load (for industries that operate only 5 or 6 days/week). The results for these two types of storage will turn out qute different, because weekend storage is not associated with an increase in collector area and the number of charge and discharge cycles is smaller than for day-night storage. The results reported below were obtained in the course of an assessment of the value of future storage technologies [Rabl and Gaul, 1981]. They were derived for the particular case of systems that provide steam at 180°C,4 but the general trends of the results have a bearing on other applications as well. The load is assumed constant (24 h/day) and the system size large enough (25 MW,) to justify using a central receiver; in addition the parabolic trough and the parabolic dish were also considered. Designs for such systems without storage have been analyzed by Bicker et al. [1981]. The results presented in this subsection have been obtained using the methods of Chapter 11, based on the collector properties of Bicker et al. [1981], As the first step, the field and receiver data of this reference were translated into a linearized efficiency equation. Table 15.4.1 lists the parameters Fmri0 and FmUfor the linearized instantaneous efficiency equation of the collector field. It also lists a term "nightloss," which expresses the losses due to cool-down at night as a percentage of total collected energy. Table 15.4.1 also states the economic assumptions. For assessing the value of future storage technologies one needs to predict both fuel costs and cost and performance of solar collectors. Both are uncertain, to say the least, and the uncertainty of the result is magnified because storage value is the difference between two uncertain terms. Under these circumstances it can be very misleading to quote a single number for the value of a future storage technology. "Assuming Tin = 176.7°Cand rout = 260"C for the collector. TABLE 15.4.1 Input for Analysis of Storage Value for Industrial Process Heat System" Collector
type
Parameters (i) Performance11 ^mlo
F m t/(W/m 2 °C) Nightloss (%) (ii) Economics c0 (K$) CA ($/m2) PWlaux [$/GJ] 7cr
Central receiver
Parabolic dish
Parabolic trough
0.86 0.22 0.54
0.87 0.42 4.60
0.77 0.39 2.50
438.00 189.00 7.73 0.169
438.00 350.00 7.73 0.169
438.00 172.00 7.73 0.169
"From Rabl and Gaul [1981]. 'Equivalent linearized efficiency equation.
442
Active Solar Collectors and Their Applications
To emphasize the sensitivity of storage value to uncertainties of the input, the calculations were carried out not only for the reference values listed in Table 15.4.1 but also for variations about these reference values. The dominant uncertainties lie in the four terms Jm Q, pfml and CA, and it seems prudent to consider + 30% variations in each of these factors. Since the first three enter only in the combination V = p^d/J^w the uncertainty of this product is \/3 X 30% = 50%. Hence we have presented the output in a 3 X 3 matrix, where the rows correspond to 0.5, 1.0, and 1.5 times the reference values of Fand the columns correspond to 0.7, 1.0, and 1.3 times the reference values of collector costs CA. The results of the value analysis are presented in Table 15.4.2 for daynight storage and in Table 15.4.3 for weekend storage. Table 15.4.2 shows TABLE 1 5.4.2 Value of Day-Night Storage [in $/MJJ for Several Collector Types and Two Locations. For Each Location and Collector Type a Matrix of 3 X 3 Values is Shown to Display the Effect of Cost and Performance Variations About the Reference Values in Table 15.4.1." Albuquerque, NM ^\
v^
Central receiver Nv^l/Cf.ref
0.5 1.0 1.5
1.0
1.3
-11 2 11
-19 -5 6
0.7
-4 7 16
Parabolic dish VI V^ "\cVQrcf 0.5 1.0 1.5
1.0
1.3
-10 3 12
-17 -3 7
0.7
-3 7 17
North-south parabolic trough V/Vrt
\ClMl.ref
1.0
1.3
-9 3 11
-16 -3 7
0.7
0.5 1.0 1.5
-3 7 16
Charleston, SC North-south parabolic trough
v^ \ C 4 / C 0.5 1.0 1.5
/l.rrf
0.7
-12 _4 3
"From Rabl and Gaul [1981].
1.0
1.3
-21 -12 -4
-30 -21 -12
443
System Optimization TABLE 1 5.4.3 Value of Weekend Storage for Two Locations for North-South Parabolic Trough. Same Format as Table 15.4.2 Albuquerque CA/CA.K*'
Y/Y«\
0.5 1.0 1.5
0.7
1.0
1.3
-1
-3 1 3
-4 -1 2
2 3
Charleston V/Vrf\
0.5 1.0 1.5
x
CA/CA.K! 0.7
1.0
-3 -1 1
-5
3 o
1.3 -7
-5 ~1
the value of storage for a process heat plant with constant load (24 h/day, 7 days/week). The first three entries are for Albuquerque, NM (A = 35°N and //, = 0.60 kW/m2) and for three collector types: parabolic trough with horizontal north-south axis, central receiver, and parabolic dish. The fourth entry is for a location with much lower insolation, Charleston, SC (X = 32.9°N, lh = 0.310 kW/m 2 ) in order to show the effect of climate. The three different collector types do not differ very much in the value they permit for storage. For the nominal case the value of storage in Albuquerque is in the range of $2-3/MJ. The effect of climate is, however, very pronounced. For Charleston the value of storage is negative except under the most optimistic assumption for performance and collector cost. Table 15.4.2 shows that storage value is extremely sensitive to cost and performance: the values range all the way from —$30/MJ (Charleston with cA/cA,nf = 1.3 and F/Fref = 0.5) to +$17/MJ (parabolic dish in Albuquerque, with cA/c4Tef = 0.7 and V/VKf = 1.5). Results for weekend storage are shown in Table 15.4.3 for a parabolic trough in Albuquerque and in Charleston. The load is 24 h/day, but only 5 days/week. For the reference case, weekend storage has a value of $1/MJ in Albuquerque and — $3/MJ in Charleston. Comparison of day-night storage with weekend storage shows weekend storage to have lower value. This is to be expected since day-night storage is useable every day, whereas weekend storage is useable roughly one-seventh of the time. The effect of collector cost reductions, while dramatic for day/night storage, disappears for weekend storage when V is large. This is a consequence of the shift from one criterion to the other in Eq. (15.4.6). These storage values are to be compared with cost projections for various storage technologies suitable for this temperature range, e.g., oil tanks, tanks
444
Active Solar Collectors and Their Applications
with oil plus rocks, and phase change salt. Tanks with oil plus rocks seem to have the lowest storage cost, on the order of $2/MJ, (Steams-Rogers Services, Inc., 1981). (The rocks also improve performance by stratification.) In sunny locations oil with rocks does look promising. REFERENCES Bergeron, K. D. 1981. "A Simple Formula for Calculating the Optimal Frequency for Cleaning Concentrating Solar Collectors." Report SAND 81-1823J. Albuquerque, NM: Sandia National Laboratories. Eason, E. D. 1978. "The Cost and Value of Washing Heliostats." Report SAND 788813. Livermore, CA: Sandia Livermore Laboratories. Eicker, P. J. et al. 1981. "Design, Cost and Performance Comparisons of Several Solar Thermal Systems for Process Heat." Reports SAND 79-8279, 8280, 8281, 8282, and 8283. Livermore, CA: Sandia Livermore Laboratories. Gordon, J. M. and Rabl, A. 1982. "Design, Analysis and Optimization of Solar Industrial Process Heat Plants Without Storage." Solar Energy 28:519. Kreider, J. F. 1979. Medium and High Temperature Solar Processes. New York: Academic Press. Kreider, J. F. and Kreith, F. 1982. Solar Heating and Cooling—Active and Passive Design, 2nd ed. New York: Hemisphere Publishing Corporation, McGraw-Hill. PURPA. 1978. Public Utilities Regulatory Policy Act. U.S. Congress. Rabl, A. and Gaul, H. W. 1981, "An Assessment of Thermal Storage Technologies for Solar Industrial Process Heat." Presented at Storage Technology Review Meeting, Golden, CO. Steams-Rogers Services, Inc. 1981. "Cost and Performance of Thermal Storage Concepts in Solar Thermal Systems—Phase I, Water/Steam, Organic Fluid and Closed Air Brayton Receivers," by Dubberly, L. J. et al., under subcontract XP-0-9001-1 for SERI. Report no. SERI/PR-XP-O-9001-l-A. Denver, CO: Steams-Rogers Services, Inc.
APPENDIX A. NOMENCLATURE
I have attempted to make the subscripts as logical and self-expanatory as possible. Also, to keep subscripts to a minimum, I have not used subscripts for quantities that refer directly to the solar collector. Therefore a few symbols differ slightly from those in other texts; for example, I have designated the U value of the collector by a simple U rather than UL. In these instances the traditional notation is indicated. Greek and special symbols are listed separately. If no units are shown in [], the quantities are dimensionless. ~d A A A^ -4dump Ag B(B) C C difc c d D / / Ji /(#) F
= = = = = = = = = = = = = = = = = =
Fm
=
F-m
=
Fx FJ.J g Gr
= = = =
levelized annual cost [$] Aap — aperture area of collector [m2] annual payment [$] absorber surface area [m2] threshold area for energy dumping [m2] ground area covered by collector array [m2] brightness distribution [W/m2 rad] or [W/m2 sterad] concentration ratio = A^/A^ capital cost of solar energy system [$] life cycle cost [$] specific heat at constant pressure [J/gm K] (also known as cp) absorber width or diameter [m] aperture width or diameter [m] solar fraction focal length [m] annual charge rate angular acceptance function 1, Fm, Fm, or FxFia depending on specification of collector temperature heat transfer factor based on mean fluid temperature Tm (also known as F and collector efficiency factor) heat transfer factor based on fluid inlet temperature T"in (also known as FR and heat removal factor) heat exchanger factor (also known as F'R/FR) radiation shape factor earth's acceleration = 9.80 m/sec2 Grasshof number 445
446
Active Solar Colectors and Their Applications
h = heat transfer coefficient [W/m2 K] ^amb — total (i.e., convective plus radiative) heat transfer coefficient from a surface to ambient [W/m2 K] ^wind = convective heat transfer coefficient from a surface to ambient [W/ m2K] H = daily total solar irradiation on collector aperture [J/m2] H(l = daily diffuse solar irradiation on horizontal [J/m2] Hh = daily hemispherical solar irradiation on horizontal [J/m2] H0 = daily extraterrestrial solar irradiation on horizontal [J/m2] i - inflation rate i = direction of incident ray / = solar irradiance [W/m2] on collector aperture Ih = beam solar irradiance at normal incidence (also called direct normal irradiance) [W/m2] 7fi = long term average of Ib during daylight hours [kW/m2] (12 h average) Id = diffuse solar irradiance on horizontal [W/m2] //, = hemispherical solar irradiance on horizontal [W/m2] /in = solar irradiance incident on aperture and accepted by absorber [W/ m2] A™ = peak solar irradiance on aperture [W/m2] /o = solar constant = 1373 W/m 2 /O.eff = effective solar constant (Eq. 2.2.1) k = thermal conductivity [W/m K] k, = hourly clearness index KT = daily clearness index = ratio of terrestrial over extraterrestrial solar radiation = Hh/H0 (also known as cloudiness index) K = extinction coefficient [m"'] K(6) - incidence angle modifer at incidence angle 0 K = all-day average incidence angle modifier L = latitude [radians] L = length [m] L = daily load [J] L - load rate [W] m = air mass m = flow rate in collector ^load = flow rate of load M = storage mass [kg] n = index of refraction n — day of year (n = 1 for 1 Jan) n = unit vector, with subscripts (e.g., c for collector normal, h for normal of horizontal surface, s for direction from earth to sun) N = lifetime for economic analysis [yr] Np = payback time [yr] A'2 = doubling time [yr] Nu = Nusselt number
Nomenclature
447
- present value [$] = energy price [$/GJ] = Prandtl number = energy [J] (daily, monthly, or yearly) = power [W] — energy per unit area [J/m2] or per unit length [J/m] = q0 — q{X + q2X2 = correlation for yearly collectible energy per aperture area [GJ/m2] as function of threshold X Qam = annual energy supplied by backup [J] q = power per unit area [W/m2] 4abs = power absorbed by collector [W/m2] 4ioSS = heat loss of collector [W/m2] R = sun-earth distance [m] r = radius of solar disk [m] r = discount rate rd = ratio of instantaneous diffuse irradiance/daily diffuse irradiance [sec~'] re = energy price escalation rate rh = ratio of instantaneous hemispherical irradiance/daily hemispherical irradiance [sec"'] r, = internal rate of return f = direction of reflected ray Ra = Rayleigh number Re = Reynolds number Sufi, = life cycle saving [$] ! = direction of transmitted ray t = solar time (i.e., time of day from solar noon) [h] (pm is +) tc = collector cutoff time [h] ts = sunset time [h] T = temperature [°C] rabs = absorber temperature [°C] Tamb = ambient temperature [°C] rcoll = collector temperature [°C] (Tabs, T-m, Tm, or rHX,,n) T"env = temperature of storage tank environment [°C] THXM - inlet temperature on load side of collector heat exchanger [°C] Tin = fluid inlet temperature [°C] ?"ioad = inlet temperature of process or load [°C] Tm = mean fluid temperature = (T-m + 7"out)/2 Toul = fluid outlet temperature [°C] ^mains = temperature of water mains [°C] rstag = stagnation temperature of collector [°C] ^stor = temperature of storage [°C] U = total heat loss coefficient of collector relative to aperture area [ W/ m 2 °C] (also known as U,) t/back = heat loss coefficient for losses through back of collector [W/m2 °C] t^from = neat loss coefficient for losses through front of collector [W/m2 °C] P pe Pr <2 Q q q(X)
448
^equiv v V X X' Xf Xf Y Yf
Active Solar Colectors and Their Applications
= = = = = = =
(— 1) X slope of linearized efficiency curve [ W/m2 °C] wind speed [m/sec] value [$] threshold = U(Tcotl - Tamb)/r,0 [kW/m2] dimensionless heat loss parameter in 0, /chart critical intensity ratio in utilizability method dimensionless heat loss parameter in /chart dimensionless insolation parameter in 0, /chart dimensionless insolation parameter in/chart
GREEK SYMBOLS a a /3 /30 &T 7 F(0) 5 A3. t 7j 7jaux ?70 ^oefr 'Jo.equiv 0 0, 0|,0i 6x: X \ v p Pground p a a a ^contour
= = = = — = = = = = = = =
absorptivity for solar radiation thermal diffusivity [m2/sec] collector tilt from horizontal [°] (+ is up) collector tilt from equatorial plane, /30 = /? — \ coefficient of thermal expansion [K~'] intercept factor end loss factor for parabolic trough at incidence angle 8 declination [°] angular radius of solar disk [mrad] emissivity efficiency efficiency of backup optical efficiency = optical efficiency including effects due to warming of glazing or reflectors = intercept of linearized efficiency curve = angle of incidence on collector [°] = solar zenith angle (angle of incidence on horizontal surface) [°] = projected angles of incidence parallel and perpendicular to collector axis [°] ~ projected indicidence angle on xz plane [°] = wavelength [nm] = latitude [°] (+ is north) = kinematic viscosity [m2/sec] = reflectance = reflectance of ground = density [g/cm3] = Stephan-Boltzmann constant = 5.670 X 10" 8 W/m 2 K 4 = rms angular half-width of radiation source or of error distribution [mrad] = optical error, with subscripts as appropriate (one-sided rms deviation from design direction) [mrad] = contour error [mrad]
Nomenclature
449
^displacement = error due to displacement of receiver relative to aperture (equivalent angular rms deviation) [mrad] o^toi = total optical error [mrad] ocular - error due to lack of perfect specularity [mrad] o-sun = rms angular width of sun [mrad] tracking T r T Tday
T,oad TOP rycar $ 0 0S 0 4> 4/ a) w, Wj
= = = = = = = = = = = — = = = =
2 [ 0"sun
i 2 ll/2 + ffoptj
tracking error [mrad] tax rate transmittance time interval or cumulative time [sec] 24 X 3600 sec = 86,400 sec annual operating time of load [sec] daily operating time of collector [sec] 365.25 X 24 X 3600 sec = 3.16 X 107 sec rim angle [°] azimuth (+ is west) [°] azimuth of sun (+ is west) [°] utilizability ground cover ratio tracking angle hour angle = 360° t/24 h sunset hour angle [°] hour angle of Eq. (2.3.18) [°]
SPECIAL SYMBOLS
£[ ]+ ^i-i
= sum over positive terms only = fraction of radiation emitted by surface / that reaches surface j, either directly or via intervening reflection(s) and/or refraction (A/P, r, N) = capital recovery factor (P/F, r, N) = present worth factor overbars = bars j3ver_insolation variables designate long term average; e.g., KT, Hh, Ib bars over economic variables indicate levelizing; e.g., pe
APPENDIX B. UNITS1
SI UNITS
SOME CONVERSIONS OF UNITS
Basic units (name, symbol, quantity)
Exact conversion factors are indicated by an asterisk(*)
meter, m length kilogram, kg mass second, s time kelvin, K thermodynamic temperature Derived units All other units are derived from basic and supplementary units. Some derived units have special names.
Length m, tn/s 1 1 1 1 1 1
ft = 0.304 8* m in = 25.4* mm mile = 1.609km ft/min = 0.00508* m/s mile/hr = 0.447 0 m/s km/hr = 0.277 78 m/s
Area m2 Decimal multiples of units The following prefixes are recommended for use with SI units: teraT 1012, gigaG 109, megaM 10", kilo k 103, milli m 10"3, micro M 10~6, nano n 10 9, pico p 1(T12, femtof 10~15, attoa 1(T18 The use of the following prefixes should be limited: hectoh 102, decada 10, decid 10~', centi c 10~2
1 ft2 = 0.092 903 04* m2 1 in.2 = 0.000645 16*m 2 1 mile2 = 2.590 km2 Volume m3, m3/kg, m3/s (Note: 1 liter = 1(T3 m3) 1 ft 3 = 28.32 liters 1 U.K. gal = 4.546 liters 1 U.S. gal = 3.785 liters 1 ft3/lb = 0.062 43 m3/kg 1 cfm = 0.471 9 liters/s 1 U.K. gpm = 0.075 77 liters/s 1 U.S. gpm = 0.063 09 liters/s 1 cfm/ft 2 = 5.080 liters/s m 2
'Reprinted from Duffie and Bcckman [1980], with permission of John Wiley & Sons.
450
451
Units
Force newton N = kg m/s2, N/m2 pascal Pa = N/m2 1 Jbf = 4.448 N 1 Ibf/ft = 14.59 N/m
1 dyne/cm = 1 (mN)/m 1 mm H2O = 9.806 65* Pa 1 bar = 105 Pa 1 psi = 6.894 kPa 1 inH 2 O = 249.1 Pa 1 mmHg = 133.3 Pa 1 at = kgf/cm2 = 98.066 5* kPa 1 atm = 101.325* kPa Energy joule J = Nm = Ws, J/kg, J/kg °C 1 kWh = 3.6* MJ 1 Btu = 1.055kJ 1 Therm = 105.5MJ 1 kcal = 4.186 8* kJ 1 Btu/lb = 2.326 kj/kg 1 Btu/lb F = 4.186 8* kJ/kg °C 1 Btu/ft2 = 0.01136MJ/m 2
1 cal/cm2 = 0.04187 MJ/m2 Power watt = J/s = N m/s, W/m2, W/m2 °C, W/m °C 1 Btu/h = 0.293 1 W 1 kcal/h = 1.163* W
1 hp = 0.745 7 kW 1 Tonrefr. = 3.51 7 kW 1 w/ft2 = 10.76 W/m2 1 Btu/h ft2 F = 5.678 W/m2 °C 1 Btu/h f t F = 1.731 W/m°C 1 Btu/h ft2 (F/in) = 0.1443 W/m °C 1 Btu/ft2h = 3.155 W/m2 Viscosity Pas = N s/m2 = kg/m s 1 cP (centipoise) = 10"3 Pa s 1 lbfh/ft 2 = 0.1724MPas Mess kg, kg/m3, kg/s, kg/s m2 1 Ib = 0.453 56* kg 1 oz = 28.35 g 1 Ib/ft3 = 16.02 kg/m2 1 g/cm3 = 103 kg/m3 1 Ib/h = 0.000 125 6 kg/s 1 Ib/h ft2 = 0.001 356 kg/s m2 Plane angle rad 2ir rad = 360* degrees Diffusivity
nf/s
1 cST (centistoke) = 10~6 m2/s 1 ft2/h = 25.81 X 10- 6 m 2 /s
APPENDIX C. PROPERTIES OF MATERIALS
Aluminum Copper Steel Stainless steel Window glass Concrete (building) Glass wool Mineral wool Polyurethane foam (rigid) Polystyrene (expanded) Ice Water Steam (at 100°C, atm. press.) Air (at 20°C, atm. press.)
k = thermal conductivity (W/m K)
p = density (kg/m3)
210 385 47.6 18 1.05 1.73 0.04-0.05 0.04 0.025 0.04 2.26 0.598 at 20°C
2675 8795 7850 7850 2500 2400 24 32 24 16 918 1000at4°C
0.025
0.60 1.204
0.026
c = specific heat at constant pressure [kJ/kg K] 0.94 at 100°C 0.39 at 100°C 0.50 — 0.84 0.84 —
2.10 4.186 1.95 1.012
Other properties Latent heat Heat of melting, ice/water Heat of vaporization, water/steam Kinematic viscosity v Air Water Thermal diffusivity a Air Water
452
333 kJ/kg 2257kJ/kgat 100°C 14.95 X 10"6 m2/sec at 20°C, atm. press. 1.01 X 10~6 m2/sec at 20°C, atm. press. 21.2 X 10~6 m2/sec at 20°C, atm. press. 0.142 X 10"6 m2/sec at 20°C, atm. press.
APPENDIX D. METEOROLOGICAL DATA
TABLE D. 1 Extraterrestrial Daily Solar Irradiation on Horizontal Surface, in MJ/m2, as Function of Latitude and Time of Year3 1 Feb. 14 Feb.
1 Mar.
15 Mar.
1 Apr.
15 Apr.
1 May
15 May
1 Jun.
15 Jun.
0.00 0.43 2.66 5.44
0.49 2.99 5.95 9.04
4.27 7.49 10.68 13.82
10.86 13.83 16.78 19.63
19.08 20.78 23.04 25.35
28.85 28.35 29.34 30.89
36.69 35.99 35.21 35.77
42.34 41.53 40.40 39.66
44.79 43.93 42.74 41.50
5.55 8,50 11.54 14.62
8.43 11.50 14.56 17.58
12.14 15.21 18.20 21.07
16.85 19.76 22.52 25.11
22.35 24.91 27.29 29.47
27.59 29.68 31.60 33.30
32.51 34.05 35.44 36.64
36.72 37.70 38.59 39.31
39.96 40.44 40.90 41.23
41.43 41.66 41.91 42.06
15.20 18.28 21.29 24.20
17.66 20.64 23.51 26.24
20.51 23.32 25.98 28.47
23.79 26.35 28.72 30.88
27.52 29.71 31.68 33.41
31.42 33.14 34.61 35.82
34.77 35.99 36.95 37.65
37.61 38.34 38.81 39.01
39.82 40.11 40.15 39.93
41.38 41.32 41.01 40.47
42.03 41.80 41.34 40.65
25.95 28.70 31.29 33.70
26.99 29.62 32.08 34.33
28.81 31.20 33.38 35.33
30.76 32.84 34.68 36.28
32.81 34.50 35.94 37.11
34.89 36.10 37.04 37.69
36.75 37.41 37.79 37.87
38.07 38.21 38.06 37.64
38.95 38.60 37.99 37.10
39.44 38.70 37.69 36.43
39.67 38.62 37.33 35.79
39.70 38.52 37.10 35.44
35.90 37.87 39.61 41.09
36.36 38.15 39.70 40.99
37.04 38.50 39.70 40.62
37.62 38.68 39.47 39.97
38.00 38.61 38.93 38.96
38.06 38.14 37.93 37.44
37.68 37.19 36.43 35.38
36.93 35.96 34.71 33.21
35.95 34.55 32.90 31.02
34.93 33.19 31.22 29.05
34.03 32.04 29.86 27.49
33.56 31.48 29.20 26.75
42.32 43.29 44.00 44.45
42.01 42.76 43.24 43.46
41.26 41.62 41.71 41.51
40.19 40.12 39.77 39.13
38.70 38.15 37.32 36.22
36.65 35.59 34.26 32.67
34.08 32.51 30.69 28.65
31.47 29.50 27.31 24.93
28.93 26.63 24.16 21.54
26.70 24.18 21.52 18.74
24.96 22.30 19.52 16.65
24.15 21.42 18.60 15.71
44.65 44.64 44.44 44.12
43.42 43.16 42.69 42.07
41.06 40.36 39.43 38.31
38.23 37.07 35.68 34.07
34.85 33.23 31.38 29.31
30.83 28.76 26.47 23.99
26.39 23.93 21.29 18.50
22.37 19.66 16.82 13.88
18.78 15.92 13.00 10.05
15.87 12.96 10.04 7.18
13.75 10.83 7.97 5.23
12.80 9.90 7.07 4.41
43.80 43.78 45.00 46.25
41.39 40.87 41.26 42.41
37.08 35.83 34.84 35.16
32.30 30.42 28.56 27.06
27.04 24.63 22.11 19.60
21.32 18.49 15.53 12.47
15.57 12.53 9.42 6.27
10.89 7.87 4.91 2.15
7.14 4.34 1.81 0.05
4.45 2.00 0.20 0.00
2.72 0.70 0.00 0.00
2.04 0.28 0.00 0.00
1 Jul.
15 Jul.
1 Aug.
15 Aug.
1 Sep.
1 5 Sep.
1 Oct.
15 Oct.
44.36 43.51 42.33 41.15
41.10 40.31 39.21 38.72
34.69 34.03 33.56 34.38
26.31 26.15 27.53 29.27
16.07 18.30 20.80 23.29
8.57 11.68 14.72 17.65
3.04 6.14 9.29 12.41
0.06 2.11 4.90 7.90
1 Jan.
15 Jan.
0.00 0.00 0.00 0.38
32
46
0.00 0.00 0.00 1.08
0.00 0,00 0.63 2.84
2.33 4.89 7.76 10.78
3.37 6.10 9.05 12.11
40 35 30 25 20 15 10 5
13.87 16.98 20.05 23.05
0
Latitude
80 75 70 65 60 55 50 45
~5
-10 -15 -20 -25 -30 -35 -40 -45 -50 -55 -60 -65 -70 -75 Latitude
80 75 70 65
1
182
16
197
213
228
60
244
75
259
91
274
106
289
121
136
1 Nov. 15 Nov.
305
320
0.00 0.00 1.66 4.21
0.000 0.00 0.13 1.93
152
167
1 Dec. 15 Dec.
335
350
0.00 0.00 0.00 0.72
0.00 0.00 0.00 0.29
453
454
Active Solar Colectors and Their Applications
TABLE D. 1 Extraterrestrial Daily Solar Irradiation on Horizontal Surface, in MJ/m2, as Function of Latitude and Time of Year3 (continued) 1 Jul. 182
15 Jul. 197
1 Aug. 213
1 5 Aug. 228
1 Sep. 244
15 Sep. 259
1 Oct. 274
15 Oct. 289
1 Nov. 305
15 Nov. 320
60 55 50 45
41.14 41.42 41.70 41.87
39.15 39.74 40.28 40.68
35.49 36.60 37.60 38.43
31.04 32.71 34.22 35.53
25.67 27.90 29.94 31.78
20.46 23.11 25.59 27.87
15.45 18.38 21.18 23.82
10.96 14.01 17.00 19.89
7.08 10.08 13.13 16.15
4.46 7.30 10.29 13.35
2.85 5.50 8.40 11.43
2.16 4.69 7.53 10.54
40 35 30 25
41.87 41.66 41.22 40.55
40.90 40.90 40.67 40.18
39.05 39.43 39.57 39.44
36.61 37.45 38.04 38.36
33.39 34.76 35.87 36.72
29.95 31.79 33.40 34.75
26.29 28.56 30.61 32.44
22.65 25.26 27.69 29.93
19.11 21.98 24.71 27.29
16.40 19.41 22.33 25.14
14.52 17.60 20.63 23.58
13.63 16.74 19.81 22.82
20 15 10 5
39.63 38.47 37.07 35.43
39.45 38.46 37.23 35.75
39.06 38.41 37.50 36.33
38.40 38.18 37.68 36.91
37.30 37.59 37.61 37.35
35.84 36.65 37.19 37.44
34.01 35.34 36.39 37.17
31.94 33.72 35.26 36.54
29.69 31.88 33.86 35.60
27.80 30.29 32.58 34.66
26.40 29.08 31.59 33.91
25.73 28.50 31.11 33.54
0 -5 -10 -15
33.57 31.50 29.24 26.81
34.04 32.12 29.98 27.66
34.92 33.26 31.38 29.29
35.87 34.58 33.04 31.26
36.80 35.99 34.90 33.55
37.41 37.10 36.50 35.63
37.66 37.88 37.80 37.44
37.54 38.27 38.71 38.87
37.08 38.31 39.27 39.94
36.51 38.12 39.47 40.56
36.01 37.89 39.52 40.90
35.77 37.77 39.54 41.05
-20 -25 -30 -35
24.23 21.51 18.70 15.82
25.18 22.55 19.80 16.97
27.01 24.56 21.95 19.23
29.27 27.07 24.68 22.13
31.96 30.12 28.07 25.81
34.48 33.07 31.42 29.52
36.79 35.87 34.68 33.22
38.74 38.33 37.64 36.67
40.34 40.45 40.29 39.84
41.37 41.91 42.18 42.18
42.01 42.86 43.45 43.77
42.32 43.32 44.07 44.56
-40 -45 -50 -55
12.91 10.01 7.19 4.51
14.08 11.18 8.31 5.55
16.40 13.52 10.62 7.74
19.44 16.64 13.75 10.83
23.36 20.74 17.98 15.11
27.40 25.07 22.55 19.86
31.51 29.57 27.41 25.04
35.44 33.96 32.24 30.31
39.14 38.18 36.99 35.60
41.92 41.42 40.70 39.82
43.85 43.70 43.36 42.88
44.80 44.83 44.68 44.42
-60 -65 -70 -75
2.13 0.33 0.00 0.00
3.01 0.90 0.00 0.00
4.98 2.46 0.46 0.00
7.90 5.06 2.43 0.35
12.15 9.14 6.13 3.22
17.02 14.05 10.99 7.84
22.49 19.77 16.93 14.00
28.19 25.93 23.59 21.32
34.06 32.44 30.93 30.09
38.84 37.91 37.43 38.34
42.37 42.07 42.88 44.08
44. 17 44.25 45.58 46.85
Latitude
a
Number under date shows day of year.
1 Dec. 15 Dec. 335 350
Figure D. 1 Monthly average clearness index KT for the United States in January (From SERI [1980]).
Figure D.2 Monthly average clearness index KT for the United States in February (From SERI [1980]).
Figure D.3 Monthly average clearness index KT for the United States in March (From SERI [1980]).
Figure D.4 Monthly average clearness index KT for the United States in April. (From SERI [1980]).
Figure D.5 Monthly average clearness index KT for the United States in May. (From SERI [1980]).
Figure D.6 Monthly average clearness index Kr for the United States in June. (From SERI [1980]).
Figure D.7 Monthly average clearness index KT for the United States in July. (From SERI [1980]).
Figure D.8 Monthly average clearness index KT for the United States in August. (From SERI [1980]).
Figure D.9 Monthly average clearness index Kr for the United States in September. (From SERI [1980]).
Figure D.10 Monthly average clearness index KT for the United States in October. (From SERI [1980]).
Figure D. 11 Monthly average clearness index KT for the United States in November. (From SERI[1980]).
Figure D. 12 Monthly average clearness index KT for the United States in December. (From SERI[1980]).
Meteorological Data
467
Figures D.I3 through D.24 depict the monthly average clearness index KT for the world from January to December (From World Meteorological Association [1980]). D.I3 January D. 14 February D.I5 March D.I6 April D. 17 May D. 18 June
D.I9 July D.20 August D.21 September D.22 October D.23 November D.24 December
Figure D. 13 January
Figure D.I4 February
Figure D. 15 March
Figure D. 16 April
Figure D. 17 May
Figure D. 18 June
Figure D.I9 July
Figure D.20 August
Figure D.21 September
Figure D.22 October
Figure D.23
November
Figure D.24 December
Figure D.25 Annual average daytime (dry bulb) temperature [°C] (From SERI [1981]).
Figure D.26 Annual heating degree days [°C day] to base 18.3°C (From SERI [1981]).
Figure D.27 Annual cooling degree days [°C day] to base 18.3°C (From SERI [1981]).
APPENDIX E. CIRCUMSOLAR DATA
TABLE E. 1 Monthly Average Circumsolar Ration Rm(X) versus Threshold Xa X(W/m2)
Yr/Mo
Hours of data
0
50
150
300
500
Atlanta, GA
77/06 77/07 77/08 77/09 77/10 77/11 77/12 78/01 78/02 78/03 78/04 78/05 78/06
163 251 273 123 246 103 73 191 234 228 135 173 216
0.081 0.063 0.109 0.076 0.036 0.039 0.049 0.057 0.055 0.043 0.068 0.050 0.071
0.078 0.059 0.100 0.071 0.034 0.033 0.044 0.051 0.049 0.040 0.064 0.045 0.066
0.070 0.051 0.083 0.058 0.031 0.029 0.034 0.045 0.044 0.037 0.056 0.033 0.061
0.057 0.041 0.060 0.043 0.027 0.020 0.022 0.034 0.033 0.030 0.050 0.023 0.050
0.039 0.026 0.039 0.024 0.019 0.013 0.018 0.022 0.023 0.025 0.039 0.018 0.031
Albuquerque, NM
76/05 76/06 76/07 76/08 76/09 76/10 76/11 76/12 77/02 77/02 77/03 77/04 77/05 77/06 77/07 77/09 77/10 77/11 77/12
80 318 199 326 106
0.021
0.020 0.028 0.026 0.031 0.039 0.039 0.024 0.035 0.037 0.045 0.058 0.037 0.035 0.053 0.040 0.039 0.040 0.057 0.052
0.019 0.025 0.024 0.028 0.037 0.035 0.023 0.033 0.031 0.040 0.051 0.033 0.031 0.050 0.036 0.036 0.039 0.054 0.049
0.016 0.022 0.017 0.025 0.027 0.030 0.022 0.029 0.023 0.036 0.039 0.028 0.026 0.042 0.031 0.031 0.037 0.048 0.042
0.015 0.018 0.014 0.020 0.019 0.021 0.017 0.024 0.017 0.028 0.027 0.022 0.022 0.031 0.024 0.027 0.032 0.034 0.036
Location
297
260 254 254 242 59 301 282 159 383 234 239 221 202
0.029 0.027 0.032 0.040 0.040 0.024 0.036 0.040 0.046 0.060 0.039 0.036 0.054 0.041 0.039 0.041 0.058 0.053
483
484
Active Solar Colectors and Their Applications
TABLE E. 1 Monthly Average Circumsolar Ration Rm(X) versus Threshold X" (continued) X(W/m2)
Location
Yr/Mo
Hours of data
Ft. Hood, TX
76/07 76/08 76/09 76/10 76/11 76/11 76/12 77/01 77/02 77/03 77/04 77/07
131 361 121 115 88 84 155 155 165 248 101 74
0.040 0.034 0.041 0.050 0.027 0.035 0.053 0.062 0.053 0.084 0.027 0.040
0.039 0.033 0.040 0.049 0.026 0.032 0.051 0.061 0.052 0.081 0.027 0.038
0.037 0.031 0.036 0.047 0.025 0.026 0.044 0.058 0.049 0.075 0.026 0.037
0.035 0.026 0.036 0.041 0.024 0.021 0.036 0.047 0.042 0.063 0.024 0.036
0.027 0.022 0.029 0.033 0.011 0.016 0.027 0.037 0.031 0.049 0.021 0.031
Argonne, IL
77/08 77/09 77/10 77/11 77/12
63 140 152 130 115
0.124 0.095 0.079 0.094 0.148
0.120 0.091 0.069 0.083 0.127
0.098 0.085 0.062 0.073 0.112
0.078 0.076 0.055 0.067 0.100
0.063 0.055 0.046 0.043 0.078
China Lake, CA
76/07 76/08 76/09 76/10 76/11 76/12 77/01 77/02 77/03
187 392 35 262 246 268 176 248 143
0.031 0.013 0.030 0.023 0.025 0.026 0.044 0.045 0.052
0.030 0.013 0.030 0.023 0.024 0.026 0.043 0.044 0.052
0.027 0.012 0.026 0.021 0.023 0.025 0.038 0.043 0.051
0.025 0.011 0.022 0.019 0.020 0.023 0.036 0.036 0.043
0.017 0.009 0.020 0.016 0.016 0.016 0.025 0.027 0.037
Barstow, CA
77/07 77/08 77/09 77/10 77/11 77/12 78/01 78/02 78/03 78/04 78/05 78/06
33 341 265 232 268 236 59 65 136 284 411 247
0.083 0.030 0.028 0.042 0.061 0.058 0.080 0.061 0.030 0.042 0.034 0.024
0.082 0.029 0.028 0.041 0.059 0.054 0.074 0.058 0.028 0.040 0.034 0.024
0.080 0.027 0.026 0.040 0.056 0.050 0.064 0.054 0.025 0.039 0.032 0.023
0.077 0.024 0.023 0.034 0.047 0.039 0.060 0.041 0.017 0.033 0.028 0.022
0.064 0.019 0.020 0.027 0.033 0.025 0.039 0.029 0.014 0.022 0.025 0.020
"From Rabl and Bendt [1982].
0
50
150
300
500
485
Circumsolar Data TABLE E.2 Standard Solar Brightness Distribution"
Break in size of angular interval
6 (mrad)b
B(0) (W/m2 sterad)c
0.218 0.654 1.091 1.527 1.963 2.400 2.836 3.272 3.709 4.145 4.581 5.018 5.454 5.890 6.327 6.763 7.199 7.636 8.072 8.508 9.381 10.690 11.999 13.308 14.617 15.926 17.235 18.544 19.853 21.162 22.471 23.780 25.089 26.398 27.707 29.016 30.325 31.634 32.943 34.252 34.561 36.870 38.179 39.488 40.797 42.106 43.415 44.724 46.033 47.342
13,631,252 13,561,148 13,441,701 13,263,133 13,012,331 12,668,456 12,185,726 11,443,798 10,061,047 7,002,494 1,730,196 144,818 63,325 51,116 43,669 36,973 31,935 27,979 24,767 21,830 17,444 13,084 10,194 8,177 6,716 5,634 4,807 4,147 3,633 3,213 2,867 2,579 2,337 2,133 1,,958 1,805 1,672 1,550 1,444 1,353 1,268 1,194 1,127 1,070 1,018 969 923 882 845 811
Solar disk
Circumsolar region
486
Active Solar Colectors and Their Applications TABLE E.2 Standard Solar Brightness Distribution3 (continued) 6 (mrad)b Break in size of angular interval
48.651 49.960 51.269 52.578 53.887 55.196
B(6) (W/m2 steradf 777 747 722 700 681 665
"From Rabl and Bendt [1982]. b Solar radius 0.275 deg (disk plus resolution of instrument). 'Brightness at angle 8 from center of sun.
487
Circumsolar Data TABLE E.3 Solar and Circumsolar Intercept Factors for Parabolic Trough with 90° Rim Angle, Cyclindrical Receiver, Concentration C, and Optical Error o-opta
C 10. 10. 10. 10.
10. 10. 10.
10. 15.
15. 15. 15. 15. 15. 15. 15. 20. 20. 20. 20. 20. 20. 20. 20. 25. 25. 25. 25. 25. 25. 25. 25. 30. 30. 30. 30. 30. 30. 30. 30. 40. 40. 40. 40. 40. 40. 40. 40.
"opt
Ts,av
3.0 4.0 5.0 6.0 8.0 10.0 15.0 20.0 3.0 4.0 5.0 6.0 8.0 10.0 15.0 20.0 3.0 4.0 5.0 6.0 8.0 10.0 15.0 20.0 3.0 4.0 5.0 6.0 80. 10.0 15.0 20.0 3.0 4.0 5.0 6.0 8.0 10.0 15.0 20.0 3.0 4.0 5.0 6.0 8.0 10.0 15.0 20.0
1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9946 0.9733 1.0000 1.0000 1.0000 0.9999 0.9989 0.9941 0.9549 0.8830 1.0000 1.0000 0.9997 0.9986 0.9905 0.9712 0.8815 0.7714 1.0000 0.9996 0.9976 0.9926 0.9696 0.9302 0.7972 0.6701 0.9998 0.9977 0.9914 0.9793 0.9369 0.8775 0.7167 0.5861 0.9960 0.9844 0.9625 0.9311 0.8504 0.7641 0.5841 0.4625
"From Rabl and Bendt [1982].
Ts,av
Tc,av
0.0262 0.0269 0.0279 0.0292 0.0324 0.0366 0.0490 0.0568 0.1031 0.1045 0.1064 0.1089 0.1149 0.1205 0.1221 0.1068 0.1787 0.1812 0.1845 0.1880 0.1929 0.1916 0.1626 0.1219 0.2432 0.2471 0.2510 0.2534 0.2501 0.2354 0.1750 0.1199 0.2990 0.3034 0.3056 0.3037 0.2866 0.2565 0.1731 0.1121 0.3903 0.3892 0.3799 0.3627 0.3132 0.2595 0.1551 0.0945
488
Active Solar Colectors and Their Applications TABLE E.4 Solar and Circumsolar Intercept Factors for Parabolic Dish of Rim Angle <j>, Flat One-Sided Receiver, Concentration C, and Optical Error aoff
30.
30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 30. 40. 40. 40. 40. 40. 40. 40. 40. 40. 40. 40. 40. 40. 40. 40. 40. 40. 40.
ff
opt
C
[mrad]
Ts,av
200. 200. 200. 200. 200. 500. 500. 500. 500. 500. 1 000. 1000. 1 000. 1 000. 1000. 2000. 2000. 2000. 2000. 2000. 5000. 5000. 5000. 5000. 5000. 10000. 10000. 10000. 10000. 10000. 200. 200. 200. 200. 200. 500. 500. 500. 500. 500. 1000. 1 000. 1000. 1 000. 1 000. 2000. 2000. 2000.
14.0 18.0 22.0 26.0 30.0 10.0 12.0 14.0 16.0 18.0 6.0 7.0 8.0 9.0 10.0 3.0 4.0 5.0 6.0 7.0 1.5 2.0 2.5 3.0 3.5 1.0 1.5 2.0 2.5 3.0 14.0 18.0 22.0 26.0 30.0 10.0 12.0 14.0 16.0 18.0 6.0 7.0 8.0 9.0 10.0 3.0 4.0 5.0
0.9539 0.8489 0.7206 0.6000 0.4984 0.9067 0.8126 0.7115 0.6163 0.5325 0.9531 0.9017 0.8372 0.7668 0.6966 0.9914 0.9525 0.8786 0.7850 0.6882 0.9874 0.9560 0.9060 0.8431 0.7737 0.9275 0.8515 0.7670 0.6810 0.5982 0.9913 0.9499 0.8711 0.7735 0.6752 0.9757 0.9305 0.8645 0.7879 0.7095 0.9911 0.9738 0.9440 0.9028 0.8535 0.9992 0.9912 0.9644
Ts,av
Tc,av
0.2467 0.2175 0.1715 0.1283 0.0943 0.3858 0.3377 0.2827 0.2310 0.1867 0.5184 0.4910 0.4511 0.4053 0.3590 0.6444 0.6294 0.5794 0.5087 0.4339 0.8029 0.7807 0.7380 0.6812 0.6171 0.8722 0.7883 0.6979 0.6078 0.5224 0.1530 0.1640 0.1537 0.1312 0.1062 0.3228 0.3124 0.2869 0.2532 0.2177 0.4422 0.4424 0.4329 0.4139 0.3878 0.5503 0.5594 0.5540
489
Circumsolar Data TABLE E.4 Solar and Circumsolar Intercept Factors for Parabolic Dish of Rim Angle , Flat One-Sided Receiver, Concentration C, and Optical Error (Topta (continued)
40. 40. 40. 40. 40. 40. 40. 40. 40. 40. 40. 40. 50. 50. 50. 50. 50. 50. 50. 50. 50. 50. 50. 50. 50. 50. 50. 50. 50. 50. 50. 50. 50. 50. 50. 50. 50. 50. 50. 50. 50. 50. 60. 60. 60. 60. 60. 60.
C
2000. 2000. 5000. 5000. 5000. 5000. 5000. 10000. 10000. 10000. 10000. 10000. 200. 200. 200. 200. 200. 500. 500. 500. 500. 500. 1000. 1000. 1 000. 1000. 1000. 2000. 2000. 2000. 2000. 2000. 5000. 5000. 5000. 5000. 5000. 10000. 10000. 10000. 10000. 10000. 200. 200. 200. 200. 200. 500.
fopi
[mrad] 6.0 7.0 1.5 2.0 2.5 3.0 3.5 1.0 1.5 2.0 2.5
3.0 14.0 18.0 22.0 26.0 30.0 10.0 12.0 14.0 16.0 18.0
6.0 7.0 8.0 9.0 10.0 3.0 4.0 5.0 6.0 7.0 1.5 2.0 2.5 3.0 3.5 1.0 1.5 2.0 2.5 3.0 14.0 18.0 22.0 26.0 30.0 10.0
7s,av
0.9145 0.8472 0.9991 0.9934 0.9779 0.9497 0.9094 0.9909 0.9627 0.9132 0.8487 0.7762 0.9961 0.9761 0.9300 0.8616 0.7817 0.9890 0.9656 0.9258 0.8724 0.8107 0.9961 0.9881 0.9730 0.9497 0.9186 0.9996 0.9961 0.9836 0.9567 0.9145 0.9995 0.9971 0.9902 0.9764 0.9542 0.9959 0.9836 0.9581 0.9181 0.8659 0.9945 0.9757 0.9394 0.8869 0.8234 0.9871
Ts,av
Tc,av
0.5277 0.4847 0.7108 0.7152 0.7108 0.6938 0.6641 0.8395 0.8185 0.7740 0.7131 0.6437 0.0931 0.1114 0.1178 0.1121 0.0992 0.2565 0.2587 0.2510 0.2345 0.2126 0.3779 0.3823 0.3823 0.3765 0.3647 0.4888 0.4973 0.5009 0.4935 0.4731 0.6477 0.6526 0.6548 0.6512 0.6399 0.7762 0.7719 0.7551 0.7233 0.6786 0.0732 0.0859 0.0921 0.0913 0.0850 0.2132
490
Active Solar Colectors and Their Applications TABLE E.4 Solar and Circumsolar Intercept Factors for Parabolic Dish of Rim Angle <j>. Flat One-Sided Receiver, Concentration C, and Optical Error
60. 60. 60. 60. 60. 60. 60. 60. 60.
60. 60. 60. 60. 60. 60. 60. 60. 60. 60. 60. 60. 60. 60. 60.
"opt
C
[mrad]
7s,av
500. 500. 500. 500. 1 000. 1 000. 1 000. 1000. 1 000. 2000. 2000. 2000. 2000. 2000. 5000. 5000. 5000. 5000. 5000. 10000. 10000. 10000. 10000. 10000.
12.0 14.0 16.0 18.0 6.0 7.0 8.0 9.0 10.0 3.0 4.0 5.0 6.0 7.0 1.5 2.0 2.5 3.0 3.5 1.0 1.5 2.0 2.5 3.0
0.9671 0.9361 0.8952 0.8469 0.9944 0.9863 0.9731 0.9545 0.9306 0.9991 0.9944 0.9821 0.9599 0.9275 0.9988 0.9952 0.9878 0.9755 0.9577 0.9925 0.9802 0.9597 0.9298 0.8905
"From Rabl and Bendt [1982].
Ts,av
Tc,av
0.2154 0.2119 0.2030 0.1896 0.3386 0.3409 0.3402 0.3360 0.3283 0.4544 0.4599 0.4608 0.4547 0.4407 0.6130 0.6153 0.6151 0.6111 0.6025 0.7355 0.7291 0.7159 0.6940 0.6629
AUTHOR INDEX
Abdel-Khalik, S. I., 227 Adams, W. J., 137, 142 Aerospace Corp., 55, 65 Alcone, J. M., 279 Anderson, B. N., 325 Andrews, J. W., 11 Antal, M. J., 5 n ASHRAE, 83 n, 88, 220, 378 ASME, 22, 1 7 5 « 3 6 0 n Atkinson, B., 373 Atterkvist, S., 8 Atwater, M. A., 59 Backus, C. E., 357, 358 Balcomb, J. D., 325 Ball, J. T., 59 Baranov, V. K., 147 Bassett, I. M., 164 Battleson, K. W., 10, 175 n Beauchamp, E. K., 373 Beckman, W. A., 28, 30, 31, 59, 120-23, 225, 226, 245-48, 261, 265, 272-74, 285,321,327-29,450 Bendt, P., 72, 111, 144, 145, 201, 202, 207, 327 Berg, R. S., 388 Bergeron, K. D., 174 Biggs, F., 136, 138, 142, 174-81, 196, 207 Boes, E. C., 64, 65, 303 n Born, M. 114 Bos, P. B., 357 Braun, J. E., 320 Brown, K. C., 421,424 Bruno, R., 286 BSE, 101 Buchberg, H., 235 Butler, B. L., 137, 141, 370, 376, 389 Butti, K., 332
Caesar, R., 373 Call, P., 368 Carlson, D. E., 357 Catton, I., 235 Cess, R. D., 128,222,237,369 Champion, R. L., 389 Chiam, H. F., 167 Claassen, R. S., 370, 376 Clark, A. F., 11, 12 Clausing, A. M., 186 Collares-Pereira, M., 18, 65-70, 103, 164, 184, 185, 309-17, 346-49 Cooper, P. I., 89 Cramer, H., 137, 142 Dellin, T. A., 174 De Meo, E. A., 357 De Winter, F., 268 Dickinson, W. C, 11, 12, 303 n, 421, 424 Dudley, V. E., 102, 300 Duffle, J. A., 28, 30, 31, 59, 120-23, 225, 226, 245-48, 261, 265, 272, 285, 321, 330, 450 Dunkle, R. V., 89 Eckart, E. R. G., 269 Edmonds, I. R., 187 Dewards, D. K., 121 n, 235, 236, 248 Eggers, G. H., 189 Eicker, P. J., 174, 175,441 Engebretson, C. D., 330 Erbs, D, G., 62, 64, 66 Faiman, D., 362 Fanney, A. H., 335 Farrington, R., 335 Felland, J. R., 235
491
492
Author Index
Fewell, M. E., 285 Francia, G., 177,235 Freese, J. M., 389-91 Fried, J. R., 381 Frohlich, C, 29, 46 Garg, H. P., 249 Gaul, H. W., 98, 112, 300, 441 Gee, R. C, 237, 240, 241, 252, 392 Gerich, J. W., 187-89 Giutronich, J. E., 156, 187 Goldstein, R. G., 226 Gordon, J. M., I l l , 292-97, 304, 305, 334, 340, 435 Govaer, D., 338 Graham, B. J., 14-15 Grandjean, N. R., 285 Grassie, S. L., 166, 167 Grether, D. F, 136,209 Grunes, H., 334 Hall, I. J., 285 Harding, G. L., 368 Harrison, T. D., 101 Hay, H. R., 332 Hay, J. E., 56, 61, 66 Hickey, J. R., 29, 47 Hildebrandt, A. F, 175 n Hill, J. E., 99, 101, 103, 105-9 Hinterberger, H., 147, 149 Hollands, K. G. T., 81, 168, 223, 224, 235 Holmes, J. T., 392 Hottel, H. C, 62-63, 81, 268, 275, 309 Howell, J. R., 128 Hunn, B. D., 51 Hunt, A., 54, 136
Kreider, J. E., 29, 44, 47, 49, 186, 215, 229, 230, 252, 260, 270, 325, 327, 364, 368, 382 Kreilh, F, 29, 44, 47, 49, 215, 221, 229-31, 252, 258, 260, 261, 270, 325, 327, 364, 368, 382 Kritchman, E., 182-84, 192 Kuehn, T. H., 226 Kurzweg, U. H., 190 Kusuda, T., 286 Kutscher, C. F., 302, 351-53, 364, 385 Lamm, L. O., 47 n Lampert, C. M., 366 Larson, D. C. 167 Lcary, P., 174 Lind, M. A., 49, 50 Lipps, F. W., 174 Liu, B. Y. H., 63, 71-72, 81, 309 Lof, G. O. G., 81, 330, 363 Lunde, P. J., 254, 284 McAdams, W. H., 220, 221 McCormick, P. G., 190 McDaniels, D. K., 167 Mclntire, W. R., 154, i 61, 162, 164 Masterson, K., 81, 112 Melnikov, G. K., 147 Mertol, A., 334, 347 Meyer, B. A., 235 Michal, C. J., 325 Mills, D. R., 149, 156, 187 Minardi, J. E., 92 n Mitchell, J. W., 305, 330 Moon, T. T., 368 NASA, 29 Nielsen, C. E., 11, 12
lannucci, J. J., 355-56 Jenkins, J. P., 103 Jones, B. W., 397 Jordan, R. C, 63, 71-72, 81, 309 Jorgenson, G., 370, 372 Karaki, S., 330 Kays, W. M., 269 Kirkpatrick, D. L., 99, 332 Klein, S. A., 81, 243, 278, 284, 285, 309, 313/z, 318, 320-21,327-29, 353 Kondratyev, K. Y., 49 Kooi, C. F, 94
O'Donnell, D. T., 358 O'Gallagher, J. J., 18, 101, 158 O'Neill, M. J., 184, 186 Owens-Illinois, Inc., 98, 99 Pearson, K. A., 353 Perlin, J., 332 Peterson, R. E., 373 Pettit, R. B., 81, 138, 141, 376, 389 Pitman, C. L., 174 Ploke, M., 147 PURPA, 429
Author Index Ratal, A., 13, 65-67, 73, 75-79, 86, 111, 116, 124, 134, 136, 148-51, 154-56, 159, 196, 210, 227, 228, 287, 291-300, 305, 309-17, 340. 374, 441, 483-90 Ramsey, J. W., 373, 221 Randall, C. M., 55, 65 Reed, K. A., 103, 132, 145 Rhee, S. J., 236 Riaz, M., 174 Ross, M, H., 5 n Russell, J. L., 189 SANDIA, 102 Schrenck, G. M., 196 Selcuk, M. K., 165 Seraphin, B. O., 368, 374 SERI, 7, 8, 57, 58, 455-66, 480-82 Sheridan, N. R., 166, 167 Siegel, R., 128 Sillman, S., 8, 325 Smestad, G., 374 Socolow, R. H., 363,411 Solar Age Magazine, 387 Solar Products Specifications Guide, 9-10 SOLMET, 53, 55, 78, 281 Sparrow, E. M, 128, 221, 222, 237, 369 Streed, E. R., 107 Sullivan, T., 380 Sunmaster, Inc. 17, 100 Sunworld Magazine, 357 Supple, R. G., 332 Tabor, H., 11, 88, 189, 218, 222, 223, 225, 231,233 Thodos, G., 268
493
USDOC, 7 U.S. Home Finance Agency, 222 Vant-Hull, L. L., 174, 194 Vitko, J., 371 Vitro Labs, 330 Vittitoe, C. N., 136, 138, 142, 174-81, 196, 207 Von Hippel, F., 76-79 Vresk, J., 368, 371-73, 377 Walton, J. D., 190 Walzel, M. D., 174 Weinberger, Z., 11 Welford, W. T., 151, 156, 192 Whillier, A., 35, 47, 268, 275, 309 Wilhelm, W. G., 11 Williams, R. H., 5 n Window, B., 164, 368 Winston, R., 26, 112, 146, 147, 149-51, 161-62, 164, 374 WMO, 58, 61,467-79 Wolf, M., 357, 358 Wood, R. L., I l l Workhoven, R. M., 102, 300
Yellot, J. I., 332 Yekutieli, G., 182-84
Zarmi, Y., 112, 303-5, 334, 338, 347, 362 Zeimer, H., 189
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SUBJECT INDEX
(A/P, r, N) definition, 401 values, 402 Absorber size approximate optimum, 171 optimization, 170 Absorptance, data, 367-68 Absorption at absorber, 122-23 atmospheric, 48-49, 174 within cover, 121-22 Absorption cooling, 331 Acceptance of diffuse radiation, 131-32 Acceptance half-angle and concentration, 125-31 of CPC, 147 definition, 84, 125 parabola, 171-72 Accuracy of tracking, data, 392-93 Acrylic, 370, 372 Air mass, 48 Air-inflated collectors, 189, 190 Albedo. See Reflectivity of ground Alzak aluminum sheet, 138, 374, 376 Ambient temperature data, U.S., 480 Amorphous silicon cells, 357 Angle of incidence equations, 40 fixed surfaces, derivation, 32-37 on horizontal, 33 tracking surfaces, derivation, 37-39 Angular acceptance function of CPC, 148 definition, 197 parabolic dish, 200-202 parabolic trough, 200 of V-trough, 165 Anisotropy of diffuse insolation, 56, 61 Annual charge rate definition, 418 effect of inflation, 397 n Annual energy. See Insolation on aperture
Annulus heat transfer equations, 225-27 results, 240-41 Antireflection coatings, 373 Aperture area, 83 n, 126 Area absorber surface, 126, 239 aperture, 83 n, 126 of collector array, 273 gross vs. net, 83 n, 103 reflector CPC, 154 parabolic, 172 ASHRAE test procedure, 88, 101, 103 accuracy, 107 specification of area, 83 Auxiliary, efficiency of, 407 Azimuth angle definition, 34 of sun, 36 Back loss, 232-34 Backup, energy form and storage, 24-25; see also Auxiliary, efficiency of Beam. See Insolation, beam Bellows, 13 Bias of data fit, 303 Biomass, 5 Black chrome, 365-67 Black fluid collectors, 92 n Black nickel, 367 Blackbody radiation, 127, 132 Blackbody spectrum, 49, 366 Blocking, 173 Bond, dust-surface, 388 Borosilicate glass, 371 Bouguer's law, 121-22, 371 Brewster angle, 119 Brightness distribution, solar/circumsolar, 133-36, 485-86 Buffer storage, 339
495
496
Subject Index
Calibrated heat source, 103-4 Capital recovery factor definition, 401 values, 402 Carnot efficiency, 359 Cassegrain optics, 193 Central heat collection, 354-56 Central limit theorem, 137 Central receiver atmospheric attenuation, 174 heat centralization, 354-56 incidence angle modifier, 174-75 optical analysis, 174-81 performance, 299-300, 441 power generation, 359, 361 power plant, 22, 175 Characteristic length, convection, 221 Charge rate, effect of inflation, 397 n Circumsolar distribution, standard, 134, 485-86 Circumsolar intercept factor, 210, 487-90 Circumsolar radiation and acceptance angle, 135-36 and collector performance, 209-11 definition, 134 Circumsolar ratio definition, 210 monthly data, 483-84 Circumsolar telescope, 54 Cleaning, 388-92 methods, 389 and reflectance, data, 390-91 Clear sky circumsolar radiation, 136 diffuse radiation, 63 radiation model, 62-63 Clearness index correlation with cloud cover, 52 correlation with sunshine hours, 52 data annual U.S., 58 annual world, 60 monthly U.S., 455-66 monthly world, 467-79 definition, 58 hourly, 64 typical values, 58 Closed loop, 106, 351-54 Cloud cover, 59 Coatings absorber, 365-69 antireflection, 373 heat mirror, 373 Cogeneration, 8, 287 photovoltaic systems, 359 Collectible radiation. See Insolation
Collector cost data, 9-10 efficiency. See Efficiency of collector efficiency factor definition, 90, 253-54 results, 254-63 mounting, 383 orientation, 381-83 technology survey, 8-23 Compound amount factor definition, 398 values, 399 Compound interest, growth rates, 405-6 Compound parabolic concentrator. See CPC Computer codes DELSOL, 174 HELIOS, 174 MIRVAL, 174 SCRAM, 174 SOLTES, 285 TRNSYS, 284-86 Computer simulation, 280-86 Concentration ratio definition, 125 and operating temperature, 132-33 of parabolic concentrators, 171-72 thermodynamic limit, 125-31 of tubular collectors, 239 Conduction back loss, 232-34 pipes, 231 Conductivity, temperature dependence, 378 Conic reflector, 190 Constant dollars, 396-97 Continuous compounding, 405-6 Contour error and reflected ray, 138-41 Controls, 277-79 Convection. See Heat transfer Conversion factors (units), 450-51 Convolution integral, 203 Cooldown rate, 108 Cooling, 330-32 Cosine loss, 175, 187 Cost avoided, 429 of collectors, 9, 10 cover materials, 370 fixed, 428 incremental, 428 reflector materials, 376-77 of saved energy comparison with other criteria, 41516
Subject Index definition, 409-10 example, 433 social, 2, 396 n of storage, 23, 444 Cover materials, 369-73 materials, data, 370 thermal effects of absorption, 246-48 transmittance, equations, 119-22 CPC angular acceptance, 148, 156 asymmetric, 149-50, 153 collector designs, 17-18 cutoff time, 43 definition, 147 design principle, 151-52 different absorber shapes, 149 for evacuated tubes, 157-64 gap, reflector-absorber, 159-64 gap loss, 159, 164 intercepted radiation, 85-86, 131-32 optical errors, 148, 156 restricted exit angles, 149, 151 n in three dimensions, 151 transmitted radiation, 154 truncation, 154-56 for tube, equation, 153 without gap loss, 161-64 Critical intensity ratio, monthly utilizability, 312 Currency, constant vs. inflating, 396-97 Current dollars, 396-97 Cutoff time, CPC, 43 Cylindrical reflector with tracking receiver, 186-88 Daily radiation. See Insolation Day number, 30 Debt financing, 407, 418-23 Declination equation, 31 graph, 35 Degree days cooling, data, U.S., 482 heating, data, U.S., 481 DELSOL (computer code), 174 Depreciation, 421, 422, 423, 424 definition, 421 effect of inflation, 424 and life cycle cost, 422, 423 Desiccant cooling, 330 Diffuse radiation. See Insolation, diffuse Direct radiation. See Insolation, beam Direct return, interconnection, 384-85
497
Dirt,388-92 Discount rate, 398-99 Diurnal variation of solar radiation, 68-69 Doubling time, 406 Dumping of energy electricity, 429-33 process heat, 341-44, 349-50 threshold for, 308 Durability, cover materials, 371-73 Dust, 388-92
Ecliptic, 29 Effective optical efficiency, 87, 247 Effective source, 196, 202-5 Effectiveness, heat exchanger, 269-70 Efficiency of auxiliary, 407 Carnot, 359 of collector conversion of temperature, 92-93 definition, 83 specification of insolation, 84-86 temperature base, 94 test procedures, 101-8 test results, 99-102 optical definition, 86 effective, 87, 247 equation, 87, 122, 205 equivalent, 89 of typical systems, 6, 331, 335, 345, 357, 361 Electric utilities, 24-25, 429 Electricity, 7, 335, 356-61 Emittance absorber, data, 367-68 of cavity, 237 effective, 214, 222 of glass, 222 Enclosure, collector, 377-78 End loss factor, parabolic trough, 98 Energy capacity cost, 416 conservation, 338 consumption industrial, 7 residential, 1 U.S., 7 world, 1 utilization, 5-8 Equation of time, 30-31 Equatorial plane, 32, 33 Equity financing, 407
498
Subject Index
Equivalent parameters for linearized efficiency, 89 Etched glass, antireflection coating, 373 Evacuated tubes designs, 13-16 optical efficiency, 239 U value, 236-40 Excess energy. See Dumping of energy Extraterrestrial. See Insolation, extraterrestrial
Fin definition, 91,264 derivation, 263-65
Fm definition, 90, 253-54 results, 254-63 ./-chart industrial process heat, 354 space heating, 327-30 water heating, 336-37 /•-number, 130 Field of view. See Acceptance half-angle Fin efficiency, 260, 365 First surface reflector, 374 Fixed reflectors with moving receiver, 18689 Flat plate collector calculation of efficiency, 241-45 description, 11 test data, 99, 100, 300 Flow balancing, 384 Flow rate measurement, 103 multi-pass systems, 327, 328 single-pass vs. multi-pass, 352-54 for specified TOM, 296 Flux concentration, 125 Focal length off-axis lens, 184-85 reflector, 117 for sagittal rays, 179 for tangential rays, 178 Fractional time distribution, 71-73 Fresnel equations, 118-19 Fresnellens, 21, 182-86 off-axis focal length, 184-85 optical design, 182-85 optical losses, 186 and tracking errors, 184 n Fresnel reflector. See also Central receiver line focus, 20 point focus, 22, 1 73-82 tracking angle, 175-76
Frost protection, 333 Fuel, 5 price escalation, 403 Gallium arsenide cells, 357 Gaussian approximation, optical analysis, 142, 206-8 Gaussian distribution, 138, 203 Getters, 16 Glass tubes, optics of, 124 Glazing. See Cover Global radiation. See Insolation, hemispherical Glycol, 379-81 Grashof number, 220 Gross area, 83 n Ground cover ratio definition, 173, 301 recommendations, 383 Ground reflectance, 51 Heat exchanger effectiveness definition, 269 results, 270 factor definition, 271 results, 272 Heat loss coefficient of collector. See U value Heat mirror, 373 Heat pipe, 13 n, 15 Heat removal factor definition, 91, 264 derivation, 263-65 Heat transfer to ambient, 217-21 approximate, 220 coefficient convectivc, typical values, 215 radiative, 214 concentrator configuration, 227-29 convection suppression, 234-35 convectivc flat surface, 220-21 honeycomb, 235 parallel plates, 222-25 tubes 221,226-27 factors, collector, 90-92 fluids, 378-81 collector test, 103 inside tubes and ducts, 229-31 radiative, plates, 222 radiative, tubes, 225 HELIOS (computer code), 174
Subject Index Heliostat cost, 10 curvature, 173, 180-81 tracking accuracy, 392 Hemispherical insolation. See Insolation, hemispherical Hemispherical reflectivity, 375 Honeycomb, 234-35 Hottel-Whillier-Bliss equation, 265, 268 Hourly clearness index, 64 Hourly radiation. See Insolation Hydraulic diameter, 229
Ideal concentrator, 147; see also CPC Imaging, 114 Incidence angle equations, 40 fixed surfaces, derivation, 32-37 modifier all-day average, 292, 312 for central receiver, 174-75 curve fits, 96, 300 definition, 96 longitudinal and transverse directions, 46 test results, 97, 98, 100, 102 trading surfaces, derivation, 37-39 Index of refraction, 117 data, 370 Industrial process heat general considerations, 338-40 multipass with storage, 351-54 no-storage system, 340-45 single-pass with storage, 345-51 temperature distribution, 7 Inflation, 396-97 effect on economic analysis, 423-24 Inlet temperature, 103, 263-65 Insolation on aperture annual, 74, 287-90 annual, above threshold, 290-99 annual, derivation, 300-305 instantaneous, 51, 301-2 monthly, 309-12 monthly, above threshold, 312-17 peak, 304, 341,432 seasonal variation, 75-78 beam data, annual U.S., 57 from hemispherical, annual, 74, 75 from hemispherical, hourly, 64-65 circumsolar. See Circumsolar radiation clear sky model, 62-63
499
diffuse accepted by collector, 131-32 clear sky, 63 data, annual U.S., 57 from hemispherical, daily, 65-66 from hemispherical, hourly, 64 from hemispherical, monthly, 67 hourly/daily, 68-69 model for anisotropy, 56, 61 extraterrestrial, daily, equation, 37, 79 daily, values, 453-54 instantaneous, 29, 30 spectrum, 48-50 frequency distribution, 71-73, 305-6 hemispherical annual data, U.S., 58 annual data, world, 60 hourly/daily, 68-69 monthly data, U.S., 455-66 monthly data, world, 467-79 instruments, 52-54 long term average, 66-71 validity of algorithm, 70 terminology, 28, 49-50 Installation, solar systems, 381-87 Insulation materials, 377-78 Intercept factor and circumsolar radiation, 209-11 definition, 205 gaussian approximation, 206-8 values, 487-90 Interconnection, collectors, 273, 384-85 Interest, present value of, 420 Interest rates, data, 397 Internal rate of return comparison, other criteria, 415-16 definition, 409 multiple solutions, 415 n Involute, 156-157 Iron, effect on transmittance of glass, 371 Irradiance. See Insolation Kinglux aluminum sheet, 374, 376 Leaks, 385 Lens, 21, 182-85 Levelizing factor definition, 403 values, 404 Life cycle cost for business, 422-23 for individual, 421-22
500
Subject Index
Life cycle (continued) and optimization, 427-28 without debt or taxes, 417-18 savings comparison, other criteria, 415-16 definition, 408 Lifetime, economic analysis, 401, 410-13 Limb darkening, 134 Line focus concentrators, image width, 142-45 Linearized efficiency equation, 88-89 Liu and Jordan correlations, 63, 64-73 Load distribution, 338 Loan annual payments, 400 effect of inflation, 397 n, 424 Long term average insolation, 66-71 Longitude, time meridians, 30
Operating temperature, photovoltaic cells, 358 Operating time, 305-7 Operation and maintenance, 3-4, 417-18 Optical efficiency. See Efficiency, optical Optical errors classification, 137 combined effect, 141-42 measurement, 111 Optimization absorber size, 170-71, 196 by computer simulation, 286 economic criteria, 415, 426-27 photovoltaic system, 429-33 process heat system, 433-36 Organic Rankine cycle, 359, 360 Outlet temperature, 103, 271 Overheating, 386-87
Maintenance, recommendations, 387 Market rates, 396-97 Materials, 364-81 absorber, 364-69 compatibility, 378, 382 cover, 369-73 heat transfer fluids, 378-81 insulation, 377-78 properties, 365-81, 452 reflector, 373-77 Microsheet, reflector material, 375, 376 MIRVAL (computer program), 174 Monthly. See Insolation Mortgage. See Loan Multiple reflections in concentrator, 116, 154,239 cover-absorber, 122-23 ground-sky, 66 within cover, 119-21
(P/F, r, N) definition, 398 values, 399 Parabola equation, 198 off-axis aberration, 197 Parabolic dish heat centralization, 354-56 power generation, 359, 361 Parabolic reflectors concentration/acceptance angle, 170-72 Parabolic trough, 10, 19 heat centralization, 354-56 power generation, 359, 361 test data, 101-2 Parasitic power, 8 n, 335, 354 Payback period comparison, other criteria, 415-16 definition, 411 and rate of return, 411-13 peak insolation, on aperture, 304, 341, 432 0,/-chart, 309, 317-22 industrial process heat, 353-54 Photovoltaics economic analysis, 429-33 power generation, 357-59 Pipe loss factors, 274-75 Pipes, 384-85 Polar tracking. See Tracking Polarization, 118 Polycarbonate, 370, 373 Power generation, 356-61 photovoltaics, 357-59 thermal, 359-61 Power tower. See Central receiver
Net area, 83 n Net present value comparison, other criteria, 415-16 definition, 408 Night loss, values, 441 Nomenclature, 445-49 Nonimaging. See CPC Nusselt Number, 215 Ocean thermal power, 359, 360 Open loop process heat system, 340-51 test procedure, 106
Subject Index Prandtl number, 220 Present worth factor definition, 398 values, 399 Principal and interest, fraction of total payment, 419-20 Prismatic reflector, 156 Profile angles, 39-43 Projected incidence angles, 39-43 Projected ray diagram, 116-17 width of sun, 143 Properties of materials, 365-81, 452 Pumps, 384-85 PURPA, utility regulation, 429 Pyranometer, 52 Pyrheliometer, 53 Radiation. See Insolation; see also Heat transfer passage, transmitted radiation, 116 shape factor, 127 Rain and cleaning, 389-91 Rate of return, internal comparison, other criteria, 415-16 definition, 409 Rates, real or market, choice for analysis, 423-24 market, 396-97 real, 396-97 Rayleigh number, 220 Real rates, 396-97 Receiver. See Absorber Reciprocity relation (radiation), 128 Reflectance data, 375, 376 of ground, 51 Reflection, law of, 115 Reflections, average number of, 116 Reflector slats fixed, 188-89 tracking, 173-77 Refraction index of, 117 law of, 117-18 Reverse return, interconnection, 384-85 Reynolds number, 221 Rim angle, 171 rms error, data fit, 303 Rock storage, 384 Rules of thumb absorber size, 171 clearness index, 58 operating temperatures, 9-11 payback time/rate of return, 410-12 seventy-year rule (doubling time), 406
501
storage value/charging cycles, 440 10% rule (annual cost), 3-4 Safety, 386-87 Sagittal rays, 177 Salvage value, 402, 422, 423 Scattering due to dust, 388 of reflected radiation, 138 SCRAM (computer code), 174 Scratches, cover, 371 Seals, 377 glass-to-metal, 13 Seasonal variation insolation on collectors, 75-78 insolation correlations, 66-67 Second law of thermodynamics, 126, 128 Second surface reflector, 374 Second-stage concentrators, 190-93 CPC, 149, 151 trumpet, 156 V-groove, 164 Selective absorber and cavity effect, 369 data, 367-68 definition, 365 Sellback of electricity, 308, 429-33 Series present worth factor, 401 Seventy-year rule, 406 Shade ring, 54 Shading, 173,381 analysis, 302 fixed surfaces, 44 Shape factor (radiation), 127 Shaped tube design, 18 test data, 101 Short circuit, thermal, 13, 268 Shorthand procedures annual insolation on aperture, 287-90 annual derivation, 300-307 utilizability, 290-99 circumsolar radiation, 289-90 general comments, 286-87 industrial process heat, 341-54 monthly, utilizability, 309-22 space heating, 327-30 water heating, 336 Side reflectors, 165-67 Silicon cells, 357, 358 Sky temperature, 218-19 and collector testing, 88 SLATS. See Reflector slats Snell's law of refraction, 117-18 Social costs, 396 n
502
Subject Index
Social discount rate, 399 Soda lime glass, 370, 371 Solar constant, 28-29 effective, 30 Solar fraction industrial process heat, 343, 351, 354 phi, f-chart, 318 space heating, 327-30 water heating, 336-37 Solar noon, 30 Solar pond deep (salt-gradient), 11-13 power generation, 359, 360 shallow, 11-12 Solar radiation. See Insolation Solar time, 30 SOLMET beam data, 55 stations, 55 typical meteorological year, 285 n weather tapes, 281 Space cooling, 330-32 Space heating performance data, 330 performance prediction, 327-30 system configurations, 326-27 Spectrum blackbody, 49, 366 solar, 48-50 standard solar, 50 Specular reflectance, 373 Specular transmittance, 371 Specularity, 138 Spherical reflectors off-axis aberrations, 177-80 with tracking receiver, 186-87 Spillover of radiation. See Intercept factor; see also End loss factor Stability of controls, 279 Stagnation, 110 Standard atmosphere, 63 Steady state and test procedure, 104-5 Stefan-Boltzmann constant in medium, 130 in vacuum, 127 Storage buffer, 339 cost, 23, 444 day-night, 339, 442 and energy use, 5-8 heat loss, 319 long term, 8, 25, 339 technologies, 23-24 value of, 439-44 weekend, 339, 443
Stratification, 24 analysis, 285 enhancement, 384 Sun, 28 angular radius, 133-36 brightness distribution, 133-36, 485-86 rms width, 135-36 spectrum, 48-50 Sunset time equation, 33 nomogram, 35 Sunshape. See Brightness distribution, solar/circumsolar Sunshine hours, 59 Support of collector, 377-78 Swimming pool heating, 337-38 System configuration, 277-78
Tangential rays, 177 Tau-alpha product. See Efficiency, optical Tax credit, 421, 422,423 Temperature ambient, data, 480 base for collector efficiency, 94 conversion, 92-93 distribution in collector, 251-52 fluid mean vs. average, 266 inlet, 91-93, 263-65 mean fluid, 90,251 stagnation, 110 typical operating, 9-10 10% rule, economic analysis, 3-4 Test procedures simplified, 108-11 standard, 101-8 Thermal network, 213 Thermal shock, 386 Thermosyphon, 334 Threshold definition, 291 for dumping or sellback, 307-8, 430 variation, 304 Tilt, effect on U value, 245 Time constant of collector, 108 equation of, 30-31 operating, 305-7 solar, 30 sunset, 33, 35 value of money, 397-98 Total energy system, 8, 287, 359 Total internal reflection, 374
Subject Index Tracking, 392-93 angle east-west axis, 39 Fresnel reflectors, 175-76 north-south axis, 38 error and collected energy, 392-93 modes, 37-39 Transients and collector efficiency, 105-6, 354 Transmittance, 119-22, 370 Transpired collector, 236 TRNSYS (computer code), 284-86 Trumpet concentrator, 156 Typical meteorological year, 285 n U value definition, 87 equivalent, linearized efficiency, 89 evacuated tubes, 236-40 flat plate collector, 242-45 parabolic troughs, 240-41 temperature dependence, 89 Units, 450-51 Utilizability annual, 290-99 definition, 290, 312 evacuated tubes, 291-92 general constraints, 304-5 monthly, 309-22
V-trough, 164-65 Value of solar system, 436 of storage, 439-44 of system changes, 437-38 Valves, 385-86 Variable volume tank, 346-47 Visibility, 63
Warmup rate, 108 Water heating, 332-38 performance data, 335 performance prediction, 336-37 system configurations, 332-34 Water mains, pressure, 347 Wind and collector testing, 88 n effect on performance, 219 effect on U value, 243-44
Yearly. See Annual; see also Insolation
Zenith Angle, 33
503
Active Solar Collectors and Their Applications
ARI RABL
OXFORD UNIVERSITY PRESS
Active Solar Collectors and Their Applications
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Oxford University Press Oxford New York Toronto Delhi Bombay Calcutta Madras Karachi Kuala Lumpur Singapore Hong Kong Tokyo Nairobi Dar es Salaam Cape Town Melbourne Auckland and associated companies in Beirut Berlin Ibadan Mexico City
Nicosia
Copyright © 1985 by Oxford University Press, Inc. Published by Oxford University Press, Inc., 200 Madison Avenue, New York, New York 10016 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, pholocopying, recording, or otherwise, without the prior permission of Oxford University Press. Library of Congress Cataloging in Publication Data Rabl, Ari. Active solar collectors and their applications. Includes index. 1. Solar collectors. TJ812R33 1985 ISBN 0-19-503546-1
Printing (last digit):
1. Title. 621.47
987654321
84-14861
